hay 2008 discrete lax pairs reductions and hierarchies (phd thesis)

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DISCRETE LAX PAIRS, REDUCTIONS AND

HIERARCHIES

by

MIKE HAY

Submitted to The University of Sydney

for the degree of Doctor of Philosophy.

Submitted August 2008



Abstract

The term ‘Lax pair’ refers to linear systems (of various types) that are related

to nonlinear equations through a compatibility condition. If a nonlinear equation

possesses a Lax pair, then the Lax pair may be used to gather information about

the behaviour of the solutions to the nonlinear equation. Conserved quantities,

asymptotics and even explicit solutions to the nonlinear equation, amongst other

information, can be calculated using a Lax pair. Importantly, the existence of a Lax

pair is a signature of integrability of the associated nonlinear equation.

While Lax pairs were originally devised in the context of continuous equations,

Lax pairs for discrete integrable systems have risen to prominence over the last three

decades or so and this thesis focuses entirely on discrete equations. Famous contin-

uous systems such as the Korteweg de Vries equation and the Painleve equations all

have integrable discrete analogues, which retrieve the original systems in the contin-

uous limit. Links between the different types of integrable systems are well known,

such as reductions from partial difference equations to ordinary difference equations.

Infinite hierarchies of integrable equations can be constructed where each equation

is related to adjacent members of the hierarchy and the order of the equations can

be increased arbitrarily.

After a literature review, the original material in this thesis is instigated by a

completeness study that finds all possible Lax pairs of a certain type, including one

for the lattice modified Korteweg de Vries equation. The lattice modified Korteweg

de Vries equation is subsequently reduced to several q-discrete Painleve equations,

and the reductions are used to form Lax pairs for those equations. The series of

reductions suggests the presence of a hierarchy of equations, where each equation

is obtained by applying a recursion relation to an earlier member of the hierarchy,

this is confirmed using expansions within the Lax pairs for the q-Painleve equa-



tions. Lastly, some explorations are included into fake Lax pairs, as well as sets of

equivalent nonlinear equations with similar Lax pairs.

To clarify the original contribution made by the author: the completeness study

of chapter 3 is based on an arχiv publication [1] of which MH is the sole author.

Chapter 4, on reductions, is based on a published collaboration [2] between MH, J.

Hietarinta, N. Joshi and F. Nijhoff, to which MH is the primary contributor. The

chapter on Hierarchies, chapter 5, is based on a publication [3] authored solely by

MH. Chapter 6 includes unpublished new material by MH.

2



Acknowledgements

I would like to thank Nalini Joshi for her thoughtful guidance throughout. Nalini is

an excellent mathematician and educator who recognizes her students’ individuality,

catering to their strengths and helping them overcome their weaknesses. It’s a rare

combination of talents.



Originality Declaration

The work in this thesis was performed between March 2004 and August 2008 in

the School of Mathematics and Statistics at the University of Sydney. The work

presented was conducted by myself unless stated otherwise. None of this material

has been presented previously for the purpose of obtaining any other degree.

Michael C. Hay

5th August 2008



Contents

1 Introduction 1

2 Background 8

2.1 Integrable ordinary difference equations . . . . . . . . . . . . . . . . . 8

2.1.1 Where they arose . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.2 QRT map and singularity confinement to construct discrete

Painleve equations . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 P∆Es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Historical setting: from continuous origins . . . . . . . . . . . 13

2.2.2 Integrable partial difference equations . . . . . . . . . . . . . . 14

2.2.3 Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Hierarchies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Completeness Study on Discrete 2×2 Lax Pairs 34

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Method of identifying potential Lax pairs . . . . . . . . . . . . . . . . 39

i



3.2.1 Links and equivalent equations at different orders . . . . . . . 42

3.2.2 Link symbolism . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2.3 Which Lax pairs need to be checked? . . . . . . . . . . . . . . 46

3.2.4 List of link combinations . . . . . . . . . . . . . . . . . . . . . 58

3.3 Derivation of higher order LSG and LMKdV systems . . . . . . . . . 59

3.3.1 LSG2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3.2 LMKdV2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.4 How most link combinations lead to bad evolution equations . . . . . 65

3.4.1 Lax pairs that yield only trivial evolution equations . . . . . . 65

3.4.2 Overdetermined Lax pairs . . . . . . . . . . . . . . . . . . . . 67

3.4.3 Underdetermined Lax pairs . . . . . . . . . . . . . . . . . . . 70

3.4.4 A special case . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4 Reductions and Lax pairs for the reduced equations 75

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2 Lax Pair and Self-Consistency of the Non-autonomous LMKdV . . . 77

4.2.1 Lax Pair of the Non-autonomous LMKdV . . . . . . . . . . . 78

4.2.2 Consistency Around a Cube . . . . . . . . . . . . . . . . . . . 79

4.3 Reductions to Ordinary Difference Equations . . . . . . . . . . . . . . 80

4.3.1 f (x) = xξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.3.2 f (x) = ¯xξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

ii



4.3.3 f (x) = ¯x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.3.4 f (x) = 1/ ¯x . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3.5 f (x) = xl+4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3.6 f (x) = 1/xl+4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.4 Lax Pairs for the Reduced Equations . . . . . . . . . . . . . . . . . . 83

4.4.1 Lax Pair for qPII . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.4.2 Lax Pair for qPIII . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.4.3 Lax Pair for qPV

. . . . . . . . . . . . . . . . . . . . . . . . . 89

4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5 Hierarchies of equations 93

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2 How to Construct the hierarchy . . . . . . . . . . . . . . . . . . . . . 94

5.2.1 Hierarchy corresponding to reductions of the type xl,m+1 =

xl+d,m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95


1/xl+d,m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.2.3 General Coefficients . . . . . . . . . . . . . . . . . . . . . . . . 102

5.3 A Known Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.4 Higher Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.4.1 Equations Corresponding to Reductions of the Type xl,m+1 =

xl+d,m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

iii



5.4.2 Equations corresponding to reductions of the type xl,m+1 =

1/xl+d,m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6 Some future directions: true and false Lax pairs and equivalent

classes of equations 115

6.1 False Lax pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.2 Equivalent evolution equations . . . . . . . . . . . . . . . . . . . . . . 122

6.2.1 Common partial difference equations . . . . . . . . . . . . . . 122

6.2.2 Higher order versions . . . . . . . . . . . . . . . . . . . . . . . 124

6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

iv



Chapter 1

Introduction

This thesis is devoted to the study of nonlinear integrable discrete equations through

their associated linear problems called Lax pairs. This broad topic is investigated

from three angles, the first aspect concerns partial difference equations (P∆Es), their

Lax pairs and how those Lax pairs are found. The second concerns reductions from

P∆Es to ordinary difference equations (O∆Es), which are the discrete analogues of

the Painleve equations, and subsequently finding Lax pairs for those O∆Es. Thethird aspect is the construction of hierarchies of O∆Es starting from expansions

within their Lax pairs. Finally, we note some results about Lax pairs that might

lead to future investigations.

Around the end of the nineteenth century, Paul Painleve (1863-1933) and his

colleagues set about classifying a large class of second-order ordinary differential

equations (ODEs). The equations studied were of the form y(t) = F (y, y , t) where

F is rational in y, algebraic in y and analytic in t, and they were classified on the

basis of whether their solutions were single-valued around all movable singularities

[4]. Those ODEs whose solutions were found to behave in this manner were said to

possess the Painleve property.

Fifty canonical classes of equations that possess the Painleve property were found

1



by Painleve [5], his student Gambier [6], and Fuchs [7]. Of the fifty, six equations

were found to define new transcendental functions, while the remainder could be

solved in terms of these six or other special functions. These six equations were thus

called the Painleve equations and are listed here:

PI: y(t) = 6y2 + t,

PII: y(t) = 2y3 + ty + α,

PIII : y(t) =1

y(y)2 − 1

ty +

1

t(αy2 + β ) + γy3 +

δ

y,

PIV: y(t) =1

2y(y)2 +

3

2y3 + 4ty2 + 2(t2 − α)y +

β

y,

PV: y(t) = 12y + 1y − 1 (y)2 − 1t y + (y − 1)2

t2 α + β y + γyt

+δy(y + 1)

y − 1,

PVI: y(t) =1

2

1

y+

1

y − 1+

1

y − t

(y)2 −

1

t+

1

t − 1+

1

t − y

y

+y(y − 1)(y − t)

t2(t − 1)2

α +

βt

y2+

γ (t − 1)

(y − 1)2+

δt(t − 1)

(y − t)2

,

where α, β , γ and δ are constant parameters and primes denote differentiation with

respect to t. Ince [8] provides derivations of the list of fifty canonical classes that

were identified by the Painleve school. ODEs may be tested for the possession of the

Painleve property with the aid of the Ablowitz-Ramani-Segur algorithm. This pro-

vides a necessary, but not sufficient, test which may be regarded as a generalization

of Frobenius analysis to nonlinear equations [9].

Integrable nonlinear equations arise as the compatibility conditions of associated

pairs of linear differential and/or difference equations often called Lax pairs [10, 11].Essential to Lax pairs associated with ODEs are the monodromy data, explained

briefly in the next few sentences. From some fixed point P on the Riemann sphere,

analytic continuation of the solution Φ around any pole of the system produces a new

solution Φ. These solutions are related by the monodromy matrix M , associated

2



with the pole in question, such that

Φ = ΦM.

The monodromy matrices define the monodromy data for the system. The com-

patibility of the type of Lax pairs that give rise to the Painleve equations ensures

that the monodromy data remains invariant through the process of analytic contin-

uation and, as such, the Painleve equations are often said to be the isomonodromy

conditions for the associated linear problems, which in turn are sometimes called

isomonodromy Lax pairs.

The monodromy data can be used to characterize the behaviour of the solutionsof the associated nonlinear system. It is possible to use the associated linear system

to solve the Painleve equation for a large class of initial data, this process is known

as the isomonodromy deformation method of solution [12]. Evidence has been put

forward that suggests a relationship between the existence of an isomonodromy Lax

pair for an equation with said equation’s possession of the Painleve property [9, 13].

Also arising as the compatibility condition of Lax pairs, nonlinear integrable par-

tial differential equations (PDEs) are intimately related to the Painleve equations.

The method used to solve PDEs using Lax pairs is rather distinct from that used to

solve ODEs, integrable nonlinear PDEs arise as isospectral conditions rather than

isomonodromy conditions. These may be solved, for a large class of initial data,

by the inverse scattering technique [14]. Well known nonlinear PDEs that have

been studied in this context include the Korteweg-de Vries (KdV) [15] and modified

Korteweg-de Vries (mKdV) [16] equations,

KdV: yt + 6yyt + yxxx = 0, (1.1)

mKdV: yt − 6y2yt + yxxx = 0, (1.2)

where the subscripts denote partial differentiation. The Painleve test has been

extended to PDEs as a test for their integrability [17]. Ablowitz, Ramani and Segur

3



observed that similarity reductions of integrable PDEs possess the Painleve property,

possibly after a change of variables [18, 19].

A natural extension of the work of the Painleve school is finding higher order or

higher degree equations that possess the Painleve property. Toward this goal, Chazy

[20], Garnier [21] and Bureau [22] began a classification of third-order ODEs of the

form y(t) = F (y, y, y , t) where F is rational in its variables. The second-order

and second-degree ODEs of the form (y(t))2 = F (y, y , t) have all been classified by

Cosgrove and Scoufis [23]. They found exactly six canonical classes of second-degree

equations that possess the Painleve property which were denoted SDI,..., SDVI.

The bustling activity surrounding integrable PDEs and Painleve equations is

motivated by their appearance in mathematical models in diverse areas of physics.

The KdV and mKdV equations describe solitary wave behaviour which arises in

atmospheric dynamics [24], ocean dynamics [25] and nonlinear optics [26]. The

Painleve equations are integrable reductions of these soliton equations and therefore

also influence the understanding of solitary waves [27, 14]. The generic solutions of

the Painleve equations are higher transcendental functions that cannot be expressed

in terms of the classical special functions [8]. The Painleve equations also arise as

models in areas such as quantum gravity [28] and general relativity [29, 30, 31].

However, this thesis is concerned with discrete equations, both P∆Es and O∆Es,

for which interest has grown rapidly since the early 1990’s, and where there continues

to be a great deal of activity today. Grammaticos, Ramani et al [32, 33] proposed a

discrete version of the Painleve property, called the singularity confinement property

(see section 2.1.2). The property has been used to derive discrete versions of the

Painleve equations, so called because they have the Painleve equations as continuum

limits. For example, a general form of the q-discrete third Painleve equation is

qPIII : xx =(x− k1κl)(x − k2κl)

(1 − k3κlx)(1 − x/k3), (1.3)

where x depends on the discrete variable l and we have use the notation x(l +1) = x,

4



x(l − 1) = x, and where κ and ki (i = 1, 2, 3) are constants.

The singularity confinement property has also been used to find integrable, non-

autonomous P∆Es. One equation thus found that is of particular interest to the

present thesis is the lattice modified Korteweg-de Vries equation (LMKdV)

LMKdV: ˆx = xx − rx

x − rx(1.4)

where we use the notation x = x(l, m + 1) and x = x(l + 1, m). Equation (1.4)

is non-autonomous because r depends on the lattice variables l and m. The test

for singularity confinement shows that the equation possesses this property when r

satisfies

ˆrr = rr.

This is solved by r = λ(l)µ(m), for arbitrary functions λ(l) and µ(m).

The names given to equations (1.3) and (1.4) arise from their continuum limits,

however, continuum limits are not unique. For example, there are at least two

well known integrable discrete equations that tend to PI, these and other discrete

Painleve equations may be found in [34]. Therefore, other forms of nomenclature

would be preferable, but these historical names persist despite the discovery of other

mechanisms that are capable of identifying the equations [35, 36].

An important method of finding and classifying nonlinear integrable O∆Es was

developed by Sakai [35]. Sakai’s methodology is based on the geometry of ratio-

nal surfaces, where each equation is defined by an affine Weyl group of Cremona

transformations on a certain family of rational surfaces obtained from P2 by blow

ups.

Another important result on classification, this time for P∆Es and by an entirely

different approach to that used by Sakai, was found by Adler, Bobenko and Suris

(ABS) in [36]. They began by equating integrability with three-dimensional consis-

tency because the latter had previously been shown to ensure the existence of a Lax

pair (and thus integrability) [37, 38], three-dimensional consistency is explained in

5



chapter 2. All three-dimensionally consistent equations on quadrilateral lattices that

met certain symmetry assumptions were classified based on a discriminant derived

from each equation. Although this represents a landmark result, and much work

has been conducted in the area since, some authors have expressed concern over

whether the assumptions used are too restrictive to capture all interesting equations

on quad-graphs [39, 40, 41].

The motivation for studying the discrete Painleve equations was initially due to

their connection with their continuous counterparts, however nowadays many now

believe that the discrete equations are actually more fundamental than the contin-

uous ones. Each continuous differential equation has many discretizations. Con-nections between difference equations and continuous equations are made through

continuum limit calculations. Conversely, many discrete equations may share the

same continuum limit (for example dPI and alt-dPI both tend to PI). Furthermore,

there exists integrable discrete equations that do not have any continuous counter-

parts. Moreover, discrete Painleve equations have appeared independently in studies

of two-dimensional quantum gravity [42, 43, 28] and orthogonal polynomial theory

[43, 44].

It has been known for some time that the singularity confinement property alone

is not sufficient for integrability [45], however many of the discrete Painleve equations

found by Grammaticos, Ramani et al are known to be integrable in the sense that

each is a compatibility condition that ensure isomonodromy for an associated linear

problem [46, 47]. The confinement property is equivalent to the well-posedness of the

discrete equation for x, even through apparent singularities on the complex sphere

of x-values in both forward and backward evolution in the lattice variables.

The outline for the remainder of this thesis runs as follows: in chapter 2, we

continue the background material on integrable nonlinear P∆Es in more detail.

Putting them into an historical setting beginning with the continuous KdV equation,

we show where the P∆Es arise, why they are considered integrable and how we can

6



find reductions from them to integrable O∆Es. A thorough review of the relevant

literature on O∆Es is also included. In the subsequent chapters, we add to the

existing body of knowledge on discrete integrable systems. A completeness study

is conducted to find all the possible 2 × 2 Lax pairs, with certain restrictions, in

chapter 3. In chapter 4 we extend a previously known method of reducing P∆Es to

O∆Es and show how the reductions can work on Lax pairs for the P∆Es to produce

Lax pairs for the O∆Es. In this way, we produce the first known 2 × 2 Lax pair

for qPIII, with multiple free parameters, amongst other results. The Lax pairs for

the discrete Painleve equations found in chapter 4 are generalized in chapter 5 and

used to find two hierarchies of q-discrete nonlinear integrable equations. Naturally,

many directions remain to be explored and major questions remain to be answered.

Some of these new directions are investigated in chapter 6, however, each chapter

ends with its own discussion outlining possible future research in the relevant areas.

7



Chapter 2

Background

This chapter summarizes part of the history of discrete Painleve equations as well as

integrable partial difference equations. Topics to be considered include: where these

equations arise, why they are believed to be discrete versions of their continuous

counterparts, and reductions from P∆Es to discrete Painleve equations. The focus

of this historical perspective lies with Lax pairs for the equations of interest, as Lax

pairs are the principal topic of this thesis.

2.1 Integrable ordinary difference equations

2.1.1 Where they arose

As pointed out in [34], from a historical perspective, Laguerre [48] was the first

to derive an integrable nonlinear ordinary difference equations. This instance was

from the perspective of orthogonal polynomials and lead to a higher order equation.

Many years later, using a similar method, Shohat [49] arrived at a second order

difference equation, which was identified as dPI decades later again. There was

another unidentified occurrence of an integrable nonlinear difference equation in

8



Jimbo and Miwa’s [50, 51, 52] papers on the continuous Painleve equations, and

they also surfaced in some field-theoretic papers. The direct link between these

difference equations and the continuous Painleve equations had to wait for a paper

by Brezin and Karakov [53] where a field-theoretic model of 2-D gravity lead to

precisely the same equation as was found by Shohat over 50 years earlier. This

time, however, a continuous limit was found to be the first Painleve equation and

so the new equation was called the discrete first Painleve equation or dPI.

The discrete Painleve equations have numerous physical applications particularly

as a partition function in quantum gravity [54, 53, 43, 28, 55].

So, nonlinear integrable difference equations have cropped up in a variety of set-

tings, even before they were identified as discrete versions of the Painleve equations.

Once they were identified as such there was greatly increased activity in their study.

The names for the equations were generally obtained from their continuous limits,

dPI tends to PI and so forth. The d in dPI indicates the type of discretization

involved with that equation. A d indicates that the discretization is of the most

common, additive type, q indicates that the discretization is of the type where the

independent variable takes on a discrete exponential form, l → ql, and no prefix

indicates that the equation is continuous. There is also the elliptic type of dis-

cretization, discovered by Sakai [35], however we do not refer to it often enough in

this thesis to require its own label.

2.1.2 QRT map and singularity confinement to construct

discrete Painleve equations

Discrete Painleve equations have been found using a number of different tech-

niques. One widely utilized method of attack was to begin with a Quispel-Roberts-

Thompson (QRT) form [56] and then apply some integrability predicting algorithm

in order to zero in on the appropriate nonautonomous terms.

9



The QRT mapping can be separated into two categories: symmetric and asym-

metric. The symmetric mapping generally has the form

xn+1 =

f 1(xn)

−xn−1f 2(xn)

f 3(xn) − xn−1f 4(xn) (2.1)

while the asymmetric form is

xn+1 =f 1(yn) − xnf 2(yn)

f 2(yn) − xnf 3(yn)(2.2a)

yn+1 =g1(xn+1) − yng2(xn+1)

g2(xn+1) − yng3(xn+1)(2.2b)

The functions f i and gi are given by the following process. Take matrices

Ai =

αi β i γ i

δi i ζ i

κi λi µi

for i = 0, 1. If the matrices Ai are symmetric then we arrive at the symmetric QRT

mapping (2.1), if not then we have the asymmetric mapping (2.2). The terms within

these matrices become the parameters in the system. Put

X =

x2

x

1

Then the functions f i and gi are obtained by

f 1

f 2

f 3

= (A0X) × (A1X)

g1

g2

g3

= (AT 0 X) × (AT

1 X)

Since the matrices Ai are 3 × 3, it would appear as though there were up to

eighteen free parameters in the asymmetric case and twelve in the symmetric case.

10



However, through various transformations, the effective number of free parameters

is reduced to eight for the asymmetric case and five for the symmetric. The system

is de-autonomized by assuming that the parameters can vary with the discrete in-

dependent variable n. So that only integrable equations are achieved, we pass the

resulting nonautonomous mapping through one of a variety of integrability detecting

algorithms to find the form of the nonautonomous terms. One of the most widely

used techniques is singularity confinement.

Singularity confinement was conceived as a discrete version of the Painleve prop-

erty. The Painleve property is possessed when the solutions to a differential equation

contain no moveable singularities other than poles. The way singularity confinementworks is to look at all the possible moveable singularities in turn and ensure that

each one is a pole in the discrete sense, i.e. that the solution returns to a finite value

within a finite number of iterations after the initial divergence.

The prototypical example of singularity confinement is dPI. We show how dPI

may be derived by requiring that a de-autonomized difference equation of the form

(2.1) possess the singularity confinement property [32]. Consider equation (2.1) with

f 1(xn) = x2n − axn − b, f 2(xn) = −xn and f 3(xn) = 0.

If a and b are now permitted to depend on n, equation (2.1) becomes

xn+1 + xn−1 = −xn + an +bnxn

. (2.3)

Equation (2.3) has a movable singularity at xn = 0. For the singularity to be

confined, an and bn must be of a specific form, we find the admissible forms below.

For initial data xn−1 = κ, xn = , it can be shown that xn−2 and xn−3 are ordinary

11



points and that

xn+1 =bn

+ O(1), xn+2 = −bn

+ O(1),

xn+3 = an+2 − an+1 + O(),

xn+4 =bn

+ O(1), xn+5 = −bn

+ O(1),

xn+6 = an+5 − an+4 + an+2 − an+1 + O(),

xn+7 =bn

+ O(1), xn+8 = −bn

+ O(1), . . . .

A first non trivial condition to yield singularity confinement or the isolated mov-

able nature of the singularity is an+2

−an+1 = 0 or an = C 1 = constant. Including

this condition in the iterations gives

xn+4 = −bn(bn+3 − bn+2 − bn+1 + bn)

(bn+2 + bn+1 − bn)+ O(1).

A second condition for singularity confinement is therefore bn+3−bn+2−bn+1+bn = 0

or bn = C 2+C 3(−1)n+C 4n. Equation (2.3) now satisfies the singularity confinement

test and is of the form

xn+1 + xn + xn−1 = C 1 + C 2 + C 3(−1)n

+ C 4nxn

. (2.4)

The first condition for singularity confinement is clearly satisfied as equation (2.4)

is linear in xn+1 and xn−1. For C 3 = 0, equation (2.4) is dPI.

It has been shown that the singularity confinement property is not sufficient to

guarantee integrability of a particular mapping and, as such, a number of refinements

have been added to the original theory including ‘keeping the memory’ and ‘non-

proliferation of pre images’.

12



2.2 P∆Es

2.2.1 Historical setting: from continuous origins

Initially put forward as a nonlinear shallow water wave model over a century ago

[57], the KdV equation

ut + uxxx − 6uux = 0 (2.5)

is now one of the most famous objects in the field of integrable nonlinear systems.

The first genus-2 solution of the KdV equation was published in 1897 by Baker in

[58]. Work on the equation laid dormant for many decades until, in 1965, Zabusky

and Kruskal published the results of their numerical experiments into the KdV equa-

tion which was found to be the continuum limit of the Fermi-Pasta-Ulam (FPU)

chain with quadratic nonlinearity [59]. There it was found that solitary wave solu-

tions, despite being nonlinear waves themselves, behaved similarly to linear waves

which obey the principal of superposition. The term ‘soliton’ was initiated to convey

the idea that such solitary waves behaved like particles.

Shortly thereafter, Muira et al. found a total of ten conserved quantities forthe KdV equation and suggested that infinitely many might exist [60, 61]. While

studying the KdV equation, Muira also examined a related entity called the modified

KdV equation (MKdV)

vt + 6v2vx + vxxx = 0 (2.6)

which is also found as a continuum limit of the FPU chain, this time with a cubic

nonlinearity. Mirroring the KdV case, Miura found another set of conservation laws

for the MKdV equation and found a connection between the two sets;

u = vx + v2 (2.7)

where solutions to the KdV and MKdV equations are represented by u and v re-

spectively. Moreover, setting

N 1(u) = ut − 6uux + uxxx = 0 (2.8)

13



and

N 2(v) = vt + 6v2vx + vxxx = 0 (2.9)

then it was shown that

N 1(u) = (2v +∂

∂x) N 2(v). (2.10)

This Miura transformation illustrates the link between the KdV and MKdV equa-

tions.

2.2.2 Integrable partial difference equations

Although Toda constructed what is now a famous, integrable differential-difference

equation [62], this was only discrete in one independent direction, and it was not

related to Lax pairs at the time. From the Lax pair perspective, the recent history

of integrable partial difference equations begins with work by Ablowitz and Ladik

in 1976 [63]. Those authors formulated an integrable difference scheme that was, in

part, motivated by a need to solve continuous PDEs numerically, but was also of the-

oretical interest. Starting with a Lax pair expansion based on the eigenvalue problem

of Zhakarov and Shabat [64], Ablowitz and Ladik derive differential-difference and

P∆E versions of the nonlinear Schrodinger equation. Solutions are also derived by

discrete inverse scattering and similar results are given in their 1977 follow up paper

[65].

Starting with an assumption about the form of a Lax pair

The method used by Ablowitz and Ladik in these early papers to obtain P∆E Lax

pairs is especially pertinent to later chapters of this thesis. The starting point was

a discrete eigenvalue problem that can be written in matrix form as follows:

θ = Lθ,

θ = Mθ.

14



Where Lax matrices L and M are given by

L =

ν b

c 1/ν

,

M =

α + 1 β

γ δ + 1

,

where ν is the spectral variable, b = b(l, m) and c = c(l, m) are lattice terms, i.e.

terms that depend only on the lattice variables l and m, and α, β , γ and δ are general

terms that may depend on the spectral variable as well as the lattice variables. The

compatibility condition for this Lax pair is LM = M L, which leads to the following

four equations:

ν (α − α) = bγ − βc, (2.11a)

1

ν (δ − δ) = cβ − bγ, (2.11b)

b − b =1

ν β − νβ + αb − δb, (2.11c)

c − c = ν γ − 1

ν γ + δc − αc. (2.11d)

At this point, directly after equation (2.14) in the original paper by Ablowitz and

Ladik, assumptions were made, based on previous results with continuous systems

[66], about the form of the general terms:

α = α0 + ν 2α2, (2.12a)

δ =1

ν 2δ2 + δ0, (2.12b)

β =

1

ν β 0 + νβ 1, (2.12c)

γ =1

ν γ 0 + νγ 1, (2.12d)

where the right hand sides are separated such that the spectral variable, ν , appears

explicitly and all those terms with a subscript depend on the lattice variables, l and

m, only.

15



Substituting (2.12) into (2.11) immediately shows that:

α2 = µ1, (2.13a)

δ2 = µ2, (2.13b)

β 0 = µ2b, (2.13c)

β 1 = µ1b, (2.13d)

γ 0 = µ2c, (2.13e)

γ 1 = µ1c, (2.13f)

(2.13g)

where µi = µi(m). Ablowitz and Ladik made also assume that µ1 = −µ2 = k,

k being a constant associated with the continuum limit of the resulting nonlinear

equation (below), and b = c, where star denotes complex conjugation. With these

values in place, it is not difficult to see that

α = k

ν 2 − 1

l j=−∞

(c( j − 1, m + 1)c( j − 2, m + 1) − c( j,m)c( j − 1, m))

,

β = k(νc

±1

ν c

),

γ = k(ν c − 1

ν c),

δ = k

1

ν 2− 1

l j=−∞

(c( j − 2, m + 1)c( j − 1, m + 1) − c( j − 1, m)c( j,m))

,

where c = c(l, m) is expressed in terms of its arguments only when needed to clarify

the summations. The resulting nonlinear equation associated with the Lax pair is

c − c = (2.14)

k(c − c − c + c)

k

c

l j=−∞

(c( j − 2, m + 1)c( j − 1, m + 1) − c( j − 1, m)c( j,m))

k

c

l j=−∞

(c( j − 1, m + 1)c( j − 2, m + 1) − c( j,m)c( j − 1, m))

.(2.15)

16



This equation is a discrete version of the nonlinear Schrodinger equation.

These equations (2.15) involve infinite sums that can be viewed as discrete ver-

sions of integrals, in this sense they are more akin to integro-differential equations

than partial differential equations such as the KdV. In any case, it should be clear

that equation (2.15) is not the most general equation that could be derived from

a Lax pair of this form. A number of assumptions were made in solving the com-

patibility condition, which effectively reduced the order and complexity of the final

equation, refer to chapter 3.2 for general examples of similar calculations. A vast

quantity of nonlinear equations have been derived in this manner, not only P∆Es,

but also PDEs, partial differential-difference equations, ODEs, O∆Es and delayequations. It is a well trodden path that is extended in chapter 3.2.

The bilinear approach

Quite a different approach was instigated by Hirota in 1977-78 [67, 68, 69] in a series

of five papers that explain how one can obtain integrable nonlinear partial differ-

ence equations through a bilinear approach. These results built on previous work on

bilinear forms in the continuous realm, by the same author. To find integrable non-

linear P∆Es, the corresponding continuous PDEs were put in to the bilinear form

by a dependent variable transformation which was discretized. The discrete bilinear

form was transformed back to a P∆E by the associated dependent variable transfor-

mation, thereby producing a discrete analogue of the original nonlinear PDE. In the

series of five papers, discrete analogues of the KdV, Toda and sine-Gordon equations

were derived along with Backlund transformations and N -soliton solutions. Similarresults for a collection of nonlinear PDEs including Burger’s equation, that were

previously known to be linearizable, were also presented.

Direct linearization of various nonlinear P∆Es has been achieved via the use of

linear integral equations having arbitrary measure and contour, based on the work

17



of Fokas and Ablowitz in the continuous case [70]. The first of such results for

discrete equations was by Date, Jimbo and Miwa in 1982 who published five related

papers [71, 72, 73, 74, 75] that uncovered many examples of P∆Es in the hope that

a taxonomy of these equations could be clarified. Their equations are arranged into

three families corresponding to Kadomstev-Petviashvili (KP), B-type Kadomstev-

Petviashvili (BKP) and elliptic equations. Further results were obtained in a similar

way by Nijhoff, Quispel et al in 1983 and 1984 [76, 56], Wiersma and Capel in 1987

[77]. Many other results of this nature were published, up to the mid 1990’s, for a

review of the topic see [78].

Consistency around a cube

Of late there has been a cacophony of activity surrounding the multidimensional

consistency approach, often referred to as consistency around a cube (CAC). The

idea is summarized, drawing largely from [40], in this section. The types of P∆Es

to which the CAC approach has so far been applied can be written as

ˆx = Q(x, x, x; plm), (2.16)

where Q(x, x, x; plm) is autonomous, first order in each of two discrete independent

variables, and are linear in each argument (affine linear). The quantity plm lists the

parameters relevant to the l-m-face. The P∆Es of interest are initially described

over two discrete dimensions l and m, to these we add a third dimension, the n

direction say, (see figure 2.1). Assume that the same equation holds along all planes

in the basic cube formed by adding the third dimension, which is to say that we

should also have the equations

˜x = Q(x, x, x; pln), (2.17)

ˆx = Q(x, x, x; pmn). (2.18)

where x = x(l,m,n + 1). Referring to figure 2.1, given the values of x, x, x and x as

initial data, located at the full circles, the values of ˆx, ˜x and ˆx, located at the open

18



x

l

m

n

x

x

x

˜x

ˆx

ˆx

ˆx

Figure 2.1: For CAC we add a third direction, n. Initial data lie at the full circles,

uniquely determined data at the open circles, and CAC requires that the there is no

conflict at the double open circle ˆx.

circles, can all be determined by Q in the relevant variables. Hence, it is possible to

calculate the value of ˆx in three different ways, corresponding to the three adjacent

faces of the cube. For the equation described by Q(x, x, x; plm) to be ‘consistent

around a cube’, the same result must be attained for ˆx in all three cases, or

ˆx = Q(x, ˜x, ˜x; plm) = Q(x, ˆx, ˆx; pln) = Q(x, ˆx, ˜x; pmn)

In 2002, Nijhoff published [38] a Lax pair for the system now known as Q4 (but

was then known as either the Adler or lattice Krichever-Novikov system), where the

Lax pair was constructed from the equation itself using its CAC property. In the

same year, Bobenko and Suris published a paper about general integrable systems

that exist on quad-graphs and possess the CAC property [37]. There, the same

method was used to construct Lax pairs for three equations that are related to oneanother: the cross-ratio equation, the shifted cross-ratio system and the hyperbolic

shifted ratio system. In the same paper, these equations were also shown to be

related to Toda type systems on the lattice. As the CAC method of Lax pair

construction is relevant to this thesis, we present the construction of a Lax pair for

LMKdV below.

19



Begin with LMKdV

ˆx = xx − rx

x − rx(2.19)

where r(l, m) = λ(l)µ(m), which has been shown to possess the CAC property [78].

As such, a Lax pair for LMKdV is constructed by allowing

x =θ2θ1

, (2.20)

where θi are the elements of the two component vector which solves the linear

systems that form the Lax pair

θ = Lθ, (2.21)

θ = Mθ.

Equation (2.21) can be written θ1

θ2

=

L1θ1 + L2θ2

L3θ1 + L4θ2

,

and taking a ratio of the components yields

θ2θ1

= ˜x =L3θ1 + L4θ2L1θ1 + L2θ2

. (2.22)

As mentioned above, the same equation must exist on all faces of the cube, so

LMKdV also arises as ˜x = q(x, x, x; pln) from equation (2.22). Comparison with

equation (2.19) shows that one satisfactory choice for the undetermined components

of the L matrix is

L1 = rx/x

L2 = 1/x

L3 = x

L4 = r

where r = λ(l)ν (n) is the product of the non-autonomous parameter functions

corresponding to the ln-plane where (2.22) exists. We therefore write the L matrix

L = D1

λx

xν

1

x

x λν

(2.23)

20



where D1 is a prefactor that does not effect equation (2.22) and is often taken to be

the square root of the determinant of L.

Because the same equation lives on the mn-plane, a similar construction shows

that the other matrix in the Lax pair is the same as L, with x replaced by x and λ(l)

replaced by µ(m). The parameter function ν (n) is regarded as the spectral variable

in the Lax pair for the equation in the lm-plane.

M = D2

µx

xν

1

x

x µν

(2.24)

It is easy to check that LMKdV arises from the compatibility condition of this Lax

pair, LM = ML, for many choices of D1 and D2, including D1 ≡ D2 ≡ 1.

The CAC approach was famously used to derive and classify all one-field equa-

tions on quad-graphs under certain assumptions [36], where the most complex equa-

tion found was the Q4 system mentioned earlier. This was a landmark paper, but

some of its assumptions have been questioned and scrutinized since. In particular

the tetrahedron assumption, which states that the expression for ˆx does not depend

on x (see figure 2.1), has received attention [40, 79]. However, at least one of theintegrable equations not possessing the tetrahedron property that have been so far,

has been shown to be solvable by direct linearization [80].

2.2.3 Reductions

Reductions from partial differential equations (PDEs) to ordinary differential equa-

tions (ODEs) provide a natural way to gain further insight into either system, de-pending on the approach. On the one hand, there is a conjectured link between the

integrability of a PDE and the possession of the Painleve property by its reductions

[19]. While on the other hand, where the reductions of some PDEs lead to Painleve

equations, the inverse scattering transform solutions to the former have been utilized

to construct solutions of the latter [18].

21



For two-dimensional continuous systems, similarity reductions involve finding

a special combination of both the independent variables such that the resulting

equation only depends on one variable, i.e. the initial PDE is reduced to an ODE.

The most famous example from the perspective of Painleve equations was first given

in [81] and reduces the modified KdV equation

ut − 6u2ux + uxxx = 0

to PII

w = 2w3 + zw + c

after usingz =

x

(3t)1/3, w(z) = u(z)(3t)1/3

and integrating once. The key lies in finding the appropriate combination of the two

independent variables, x and t, which is done using symmetry arguments about the

initial PDE.

Discrete symmetry reductions

The focus of this thesis is entirely on discrete equations, which presented their own

problems, distinct from those of their continuous analogues. The study of reductions

of partial difference equations to ordinary difference equations was conducted most

vigorously by Nijhoff and collaborators in the 1990’s. In these works [82, 83] the

analogue of a similarity reduction in a discrete setting was given to be a pair of

equations, termed the lattice equation and the similarity constraint, defined such

that localized configurations of initial data can be iterated throughout the lattice

leading to a global solution. The example of this type of reduction most pertinent

here is that which reduces the LMKdV equation to a discrete form of the second

Painleve equation. The form of the LMKdV equation used is

ˆx = xx − rx

x − rx(2.25)

22



where r is an autonomous parameter. If we consider which lattice points, (l, m), are

involved at some iteration of the LMKdV equation (2.25), we arrive at the plaquette

of figure 2.2, where the axes are included to indicate directions only, the plaquette

(l, m)

(l, m + 1)

l + 1, m)

(l + 1, m + 1)

l

m

Figure 2.2: A square plaquette with four vertices.

drawn might lie anywhere in the (l, m) plane. Since LMKdV (2.25) is linear in each

iteration of x, the value of x, at any vertex of the plaquette in figure 2.2, can be

written in terms of the values of x at the other three vertices.

Turning to the similarity constraint

lx− x

x + x+ m

x − x

x + x

=3k1 − 1

2− k2(−1)l+m, (2.26)

where x is the dependent variable and ki are constants, we see that four vertices are

involved which lie in the arrangement shown in figure 2.3.

Equations (2.25) and (2.26) are iterated to form the global solution, starting

with a configuration of initial data of the form shown in figure 2.4. The value of xat some locations is found using the lattice equation (2.25), these are marked with

stars , while the value of x at other locations is found using the similarity constraint

(2.26), these points are marked with circles ◦. At some location there will be two

expressions for x, one from each of the lattice equation and the similarity constraint,

this point is marked with the symbol . The two expressions at the point marked

23



(l − 1, m) (l + 1, m)

(l, m − 1)

(l, m + 1)

l

m

Figure 2.3: A cross-shaped plaquette with four vertices.

(l − 1, m + 1)(l, m + 1)

(l, m) (l + 1, m) l

m

Figure 2.4: Initial data occupying four vertices in a stair configuration.

in figure 2.5 must be equivalent for the lattice equation and similarity constraint

to be compatible. It has been put forward that this type of compatibility could be

used as the definition of symmetry for discrete equations [84].

In some cases, compatible pairs of lattice equations and similarity constraints

can be used to completely remove the dependence on one of the lattice variables

and thereby reduce the lattice equations, that are P∆Es, to O∆Es. The example of

LMKdV (2.25) with the similarity constraint (2.26) given above, reduces LMKdV

to a rich O∆E that can be considered to be a discrete form of P II, PIII , PV or PVI

24



◦

◦

◦

◦

Figure 2.5: Part of the global solution generated by either the lattice equation (2.25)

marked

◦, or the similarity constraint (2.26) marked , the position marked has

two expressions, one from both the lattice equation and the similarity constraint.

Full circles indicate initial data positions.

as follows [82, 85]: define a and b to represent the terms on the left hand side of

equation (2.26).

a =x − x

x + x(2.27)

b =x − x

x + x

. (2.28)

Also define u and v as

u =x

ˆx(2.29a)

v =x

x. (2.29b)

The key is to remove the dependence of a and b on either of the lattice variables l or

m, we choose to remove m here. Notice that a is already without terms that have

been shifted in m, we can write a in terms of u and v quite simply

a =v − u

v + u. (2.30)

An expression for b that only involves l shifts can be obtained by taking equation

(2.29) and adding its backward m-shifted counterpart to either side. Rearranging

25



yields

(u + r)b + u = (v − r)b + v. (2.31)

The similarity constraint (2.26) is used to write b as

b =1

m

k1 − k2(−1)l+m − l

v − u

v + u

(2.32)

which can be substituted into equation (2.31) to obtain the sought after nonlinear

O∆E

(l + 1)(r + x)(1 + rx)x − x + r(1 − xx)

x + x + r(1 + xx)− n(1 − r2)x

x − x + r(1 − xx)

x + x + r(1 + xx)

= k1r(1 + 2rx + x2) + k2(−1)l+m(r + 2x + rx2) −mr(1 − x2)

A type of reduction for nonautonomous partial difference equations

In [86] Grammaticos, Ramani and collaborators published a new method by which

to construct discrete Painleve equations starting with partial difference equations

(P∆Es). An important point of difference between this and previous work on the

subject was that, in the new setting, the P∆Es used were non-autonomous, where

previous studies on discrete reductions had used only autonomous equations. For

example, one P∆E studied was a non-autonomous version of the LMKdV (2.25) en-

countered in the previous section. This equation was made autonomous by allowing

the parameters k and q to depend on the lattice variables l and m. The dependence

is not arbitrary, a special dependence was found using singularity confinement to

retain integrability of the equation. The form of LMKdV used in [86] was

ˆx = xrx− x

rx− x(2.33)

where the non-autonomous term r must be separable, i.e. r(l, m) = λ(l)µ(m), to

ensure its integrability. The condition on r, derived by singularity confinement, is

ˆr = rr (2.34)

which plays an important role later on, and is clearly solved when r is separable.

26



Two reductions from LMKdV to q-discrete Painleve equations were presented.

The first involved a reduction of the form

x = ¯x. (2.35)

Simply substituting equation (2.35) into (2.33) and introducing y = ¯x/x results in

the following nonlinear O∆E

yy =1 − ry

y(r − y). (2.36)

The form of the equation is tantalizingly similar to some known q-discrete Painleve

equations, however, it cannot be identified until the non-autonomous terms are

known explicitly, which requires further analysis.

On the (l, m) lattice, this reduction is described by figure 2.6. The reduction

m

l

Figure 2.6: Lines joining lattice points of equal x value under the reduction x = ¯x.

used to achieve equation (2.35) was carried out on the plaquette at lattice points

(l, m), (l + 1, m), (l, m + 1) and (l + 1, m + 1), and the reduced equation can be

thought to exist along the points where m is fixed in the (l, m) lattice, i.e. along

the l axis in figure 2.6. However, the same reduction should work anywhere in the

(l, m) plane and we can find the explicit form of the non-autonomous terms in the

reduced equation by considering the P∆E at a position moved up one step in m. A

27



’hatted’ version of equation (2.33) is

ˆx = xr ˆx − ˆx

r ˆx − ˆx. (2.37)

Noting that, under the present reduction, ˆx = ˆx = 4x, equation (2.37) is reduced to

¯yy =1 − r ¯y

¯y(r − ¯y). (2.38)

from where we may shift the whole equation down twice in the l direction to obtain

yy =1 − ry

y(r − y). (2.39)

A comparison between equations (2.36) and (2.39) shows that the condition that r

must satisfy is the same as the reduction on x

r = ¯r. (2.40)

The explicit form of r is gleaned when this condition is used in conjunction with

equation (2.34)

r = k1kl2k(−1)l3 (2.41)

where ki are constants. With r of this form, equation (2.36) is identified with qPII

when k3 = 0 [87], or with an asymmetric form of qth when k3 = 0 [88].

The second reduction from LMKdV, reported in the same paper, used x = ¯x

and resulted in

ww =1 − rw

r − w(2.42)

where w = ¯x/x and log r = k1 + k2l + k3 j3l + k4 j32l with j33 = 1. This equation was

identified as qPII when k3 = k4 = 0 [87], or as qf i in the generic case.

Grammaticos et al. also documented two reductions of this type from the lattice

sine-Gordon equation (LSG) to q-discrete Painleve equations, all working in the

same way as described above, and using LSG as their starting point.

ˆx = x1 + rxx

xx + r. (2.43)

The following reductions hold:

28



• x = x: leads to xx = 1+rxr+x

, where r = k1kl2, this is a special case of qPIII .

• x = ¯x: leads to yy = y(1−ry)r−y , where y = ¯x/x and log r = k1 + k2l + k3(−1)l,

this equation is equivalent to (2.36) on taking the reciprocal of either even orodd iterates of y.

Other reductions of this kind are also possible, see chapter 4. The series of

reductions presented in chapter 4 suggests the existence of hierarchies of integrable

q-difference equations, this is confirmed in chapter 5.

2.3 Hierarchies

This section contains a summary of the literature that pertains to chapter 5, where

the construction of two new hierarchies of q-discrete O∆Es is explained, based on

Lax pair expansions.

The first “higher order” integrable system of Painleve type was published by

Garnier in 1912 [21] and is now known as the Garnier system. These are systems

arising from compatibility of linear systems with n singularities, where n ≥ 4 (while

the linear system for PVI has three singularities). The resulting compatibility con-

ditions are nonlinear PDEs with n − 2 independent variables, or ODEs with one

independent variable and n − 3 parameters. They provide hierarchies of arbitrary

order [21, 89]. Since then, the majority of results concerning higher order integrable

equations have been on partial differential and/or difference equations.

Most results in the literature on hierarchies of discrete equations have concen-

trated on partial difference or differential-difference equations. The earlier work,

since around the beginning of the 1980’s, concentrated on the Toda lattice hierar-

chy, as well as its variations such as the relativistic Toda Hierarchy [90, 91, 92, 93].

Other lattice hierarchies that have since been studied include a q-discrete version

29



of the KP hierarchy [94, 95, 96], and generalizations thereof such as the ‘universal

character’ hierarchy [97], or the gl3 hierarchy [98].

While these have led to some reductions to additive O∆Es [89, 95, 98, 97], no

previous attempts appear to have been made to find hierarchies of q- difference

equations via reductions. In chapter 5, we find hierarchies of q-difference equations

through the expansion of Lax pairs. Those expansions are motivated by two series

of reductions from a single P∆E, LMKdV, but those reductions play no part in the

actual formation of the hierarchy.

Hierarchies via Lax pair expansions

The publications most relevant to the results presented in chapter 5 are [99, 100]. In

those papers hierarchies of d-discrete Painleve equations are derived from expansions

within differential-discrete Lax pairs. For example, the dPII hierarchy derived in [99]

begins with the following linear problem

θ = Lθ, (2.44a)

∂θ∂ν

= N θ, (2.44b)

which is a differential-difference isomonodromy Lax pair with spectral variable ν . L

and M take the forms

L =

ν x

x 1/ν

, (2.45a)

M = A B

C −A , (2.45b)

where x = x(l), l being the discrete, independent variable and A, B and C depend

on l and ν . The compatibility condition for (2.44) is

∂L

∂ν + LM = M L (2.46)

30



which gives three equations for A, B and C in terms of x and ν . B and C can be

eliminated from the system, resulting in an expression for A

ν xx + xx A(1 + x2

) + A(x2

− 1)− xx A(1 + x2

) + A(x2

− 1)+ν 2xx(A− A) − x

ν (x + x) − xx

ν 2(A− A) +

xx

ν 3= 0. (2.47)

It is convenient to rewrite this in the following operator form

ν − x

ν (

1¯x

+1

x) =

(ν 2 +

1

ν 2)(∆2 + ∆) + J

A, . (2.48)

where

J = x ∆(∆ + 2)((x −1

x)∆ + 2x) −2

x∆ . (2.49)

The hierarchy is brought about by allowing A to be an expansion in powers

of ν , with coefficients that depend on l, to be determined from the compatibility

condition. Recalling that every order of ν that arises in (2.47) forms a separate

condition, equation (2.47) suggests that A be rational in ν . As such, we allow A to

adopt the following form

A = a1ν

+

mi=1

(ν (2i−1) + 1ν (2i+1)

)a2i+1, (2.50)

where ai = ai(l). Using this expansion in equation (2.48) results in a set of equations

for the terms ai, each equation multiplied by a distinct order of the spectral variable

ν . In fact, there are twice as many distinct orders of ν as there are variables ai,

however, each equation repeats twice, resulting in precisely the correct number of

conditions to solve for every ai in the expansion, plus one extra condition that

becomes the evolution equation. At the extreme orders, ν (−2m−3) and ν (2m+1), we

find

(∆2 + ∆)a2m+1 = 0

which is solved for a2m+1 ≡ k2m+1, with k2m+1 a constant. At the next orders in

31



from the extremes, ν (−2m−1) and ν (2m−1), we make the following calculation

J a2m+1 + (∆2 + ∆)a2m−1 = x∆(∆ + 2)2xk2m+1 + (∆2 + ∆)a2m−1,

= 2k2m+1x(¯x− x) + ¯a2m−1 − ¯a2m−1,= 2k2m+1∆(xx) + ∆ ¯a2m−1,

= 0,

which is solved for a2m−1 = −2k2m+1xx + km+1, where km+1 is another constant.

Moving successively from these extreme orders in ν toward the inner orders, we

obtain

a2i−3 = −a2i+1 − (∆2

+ ∆)−1

a2i−1 for i = 3, . . . , m , (2.51)

a1 = l − a5 − (∆2 + ∆)−1a3, (2.52)

x

1¯x

+1

x

= −2(∆2 + ∆)a3 − J a1. (2.53)

Equation (2.51) is used iteratively to calculate the coefficients a2m−3 to a3 one by one,

equation (2.52) gives the value of a1 and the final equation, (2.53), is the evolution

equation of the system. Thus, each positive integer value of m yields a different

equation in the hierarchy, m = 0 delivers a trivial equation, m = 1 brings about thewell know (second order) dPII equation:

x + x =(k1 + k2l)x + k3 + k4(−1)l

1 − x2

where ki are constants, manipulated to retrieve the familiar form of dPII. The next

equation in the hierarchy is fourth order, obtained when m = 2

2k4(¯x(1 − x2) − x(1 − x2))(1 − x2) − 2k4x(x + x)2(1 − x2)

+2k3(x + x)(1 − x2) + x(2k2 + 2l + 1) = k1 + k0(−1)l

As m is increased, evolution equations are produced that are of order 2 m and are

of increasing complexity.

No general formula for the mth equation in the hierarchy is given in [99], one

must calculate each member in turn. Although, it is not difficult to produce the

32



m + 1st member from the mth because all of the coefficients ai, i > 3, remain the

same, except for a change of index ai → ai+2. The new coefficients a1 and a3 are

calculated using (2.51) and (2.52), then the new evolution equation is (2.53). The

hierarchies produced in chapter 5 do not contain any equivalent of ai that remain

unchanged from one member of the hierarchy to the next. However, a general

formula for every coefficient is given, as is the form of every equation in the hierarchy.

It is interesting to note that, while the Lax pairs for successive equations in that

hierarchy become increasingly complex with increasing order of the equations, the

equations themselves retain the same simple form while their order increases.

33



Chapter 3

Completeness Study on Discrete

2×2 Lax Pairs

3.1 Introduction

Despite the existence of a Lax pair often being used as the definition of integrabil-

ity for a given equation [101], there have been few studies that sought to find or

categorize nonlinear equations that used Lax pairs as their starting point. Of those

studies that did begin with Lax pairs, most chose a form of the Lax pair a priori ,

that is an assumption was made concerning the dependence of the linear systems

on the spectral parameter, thus limiting the possible results.

Lax pairs can appear in many guises, the type that we are exclusively concerned

with in this chapter consist of a pair of linear problems written:

θ(l + 1, m) = L(l, m)θ(l, m)

θ(l, m + 1) = M (l, m)θ(l, m).(3.1)

where θ(l, m) is a two-component vector and L(l, m) and M (l, m) are 2×2 matrices.

These linear problems are described by L and M , which are referred to as the Lax

34



matrices. The easily derived compatibility condition on this Lax pair is

L(l, m + 1)M (l, m) = M (l + 1, m)L(l, m)

and it is through this compatibility condition that we arrive at the integrable non-

linear equation associated with the Lax pair.

The present chapter is focussed wholly on Lax pairs that are 2×2, where each

entry of the Lax matrices contains only one separable term. The Lax pairs are

otherwise general in that no assumptions are made as to the explicit dependence

of any quantities within the Lax matrices on the lattice variables, l and m, or on

the spectral variable n. By one separable term we mean that each entry contains aterm that can be split into a product of two parts, one that depends on the lattice

variables, and another that depends only on the spectral variable. For example, in

the 11 entry of the L matrix, we write the separable term a(l, m)A(n). Both a and

A may contain multiple terms themselves, say a =

i ai, A =

j A j, however, all

terms within a must multiply all those within A, the other entries their own similar

term. So, we could have an L matrix

L = [a1(l, m) + . . . aM (l, m)][A1(n) + . . . + AN (n)] b(l, m)B(n)

c(l, m)C (n) d(l, m)D(n)

where we have written the 11 entry in the expanded form and left the other entries

abbreviated to save space, no other terms can be added into any entry. M must also

contain just one separable term in each entry, although these terms are independent

of those in L.

The reason for limiting the Lax pairs to those that are 2 × 2 with one separableterm in each entry is two fold. Firstly, Lax pairs with more terms typically lead

to equations of higher order, as can be seen from hierarchies of equations with

Lax pairs [3, 99], therefore we constrain our study to the lower order equations

by limiting the number of terms. Secondly, we limit the number of terms present

in the compatibility condition and thus render it less complicated to examine all

35



of the combinations of terms that can arise there. A combination of terms in the

compatibility condition defines a system of equations that we subsequently solve, in

a manner that preserves its full generality, up to a point where a nonlinear evolution

equation is apparent, or it has been shown that the system cannot be associated

with a nonlinear equation. Testing all combinations of terms, we thereby survey the

complete set of Lax pairs of the type described.

In fact, of all the potential Lax pairs identified by this method, only two lead

to interesting evolution equations. These are higher order varieties of the lattice

sine-Gordon (LSG) and the LMKdV equations, which can be found in section 3.1.1.

The remaining systems are shown to be trivial, overdetermined or underdetermined.This finding adds impetus to the suggested connection between the singularity con-

finement method and the existence of a Lax pair made in [102].

As we do not make any assumptions about the explicit dependence on the spec-

tral parameter, we show that a particular nonlinear equation may have many Lax

pairs, all depending on the spectral parameter in different ways. The effect that this

freedom has on the process of inverse scattering is, as yet, unclear.

This chapter is organized as follows: section 3.1.1 presents the major results,

those being the higher order versions of LMKdV and LSG, as well as a statement

of the completeness theorem. The method of identifying and analyzing the viable

Lax pairs is laid out in section 3.2, where a representative list of all the Lax pairs

identified can be found. Section 3.3 explains how the higher order LSG and LMKdV

equations are derived from the general form of their Lax pairs and section 3.4 pro-

vides examples that describe why most Lax pairs found in section 3.2 lead to trivial

systems. A discussion section rounds out the chapter.

36



3.1.1 Results

Note that all difference equations in the remainder of this chapter will use the

notation

x = x(l + 1, m),

x = x(l, m + 1).

As one of the two main results of this chapter, we present two new integrable

nonlinear partial difference equations. The first equation,

LSG2:

ρ

σ

x

x+ λ

1µ1

ˆxy =σ

ρ

ˆx

x+

λ2µ2

xy(3.2a)

σ

ρ

ˆy

y+

λ2µ2

xy=

ρ

σ

y

y+ λ1µ1xˆy (3.2b)

is referred to as LSG2 because it is second order in each of the lattice dimensions,

and because setting x = y returns the familiar LSG equation, (3.3) below, in a non-

autonomous form. Here x = x(l, m) and y = y(l, m) are the dependent variables,

λi = λi(l) and µi = µi(m) are functions of the lattice variables that play the same

role as parameters in autonomous P∆Es, as are ρ = λ(−1)m

3and σ = µ(−1)l

3.

xˆx =λ2µ2 − ρ

σxx

λ1µ1xx− σρ

(3.3)

Similarly, the equation

LMKdV2:λ1

σ

ˆx

x+

µ2

ρ

y

y= λ2σ

y

y+ ρµ1

ˆx

x(3.4a)

ρµ1xˆy + λ2σxy =µ2

ρxy +

λ1

σxˆy (3.4b)

where the terms are as for LSG2, is referred to as LMKdV2, again because setting

x = y brings about LMKdV as in equation (3.5) below. Note that the version of

LMKdV so attained is of a more general form than the most common variety listed

in chapter 2, equation (2.33).

ˆx = xλ2σx− µ2

ρx

λ1σ

x − µ1ρx(3.5)

37



The following definition is introduced to clarify Theorem 3.1 below:

Definition 1 A separable term is one that can be written as a product of quan-

tities, one quantity that depends solely on the lattice variables and another that

depends solely on the spectral variable.

Note, as described in the introduction to this chapter, each of these two quantities

that make up a separable term can possibly be expressed as a sum, provided that

all parts of the sum depending on the lattice variables multiply all those depending

on the spectral variable.

The second main result is the following theorem:

Theorem 3.1 The system of equations that arise via the compatibility condition of

any 2 × 2 Lax pair (3.1) with one nonzero, separable term in each entry of each

matrix is either trivial, underdetermined, overdetermined, or can be reduced to one

of LSG 2 or LMKdV 2.

The proof of Theorem 3.1 lies in considering all of the possible sets of equations

that can arise from the compatibility condition of such Lax pairs, and solving those

sets of equations in a way that retains their full freedom. This proof occupies the

remainder of the chapter.

The Lax pairs associated with LSG2 and LMKdV2 respectively are listed below,

these Lax pairs are derived in section 3.3. The Lax pair for LSG2 is

L = F 1/ρ F 2λ1x

F 2λ2/x F 1ρy/y

(3.6a)

M =

F 2x/(σx) F 1/y

F 1µ1y F 2σ

(3.6b)

38



While the Lax pair for LMKdV2 is

L =

F 1λ1x/x F 2ρ/y

F 2y/ρ F 1λ2

(3.7a)

M =

F 1µ1x/x F 2y/σ

F 2σy F 1µ2

(3.7b)

where F i = F i(n) are arbitrary functions of the spectral variable n with the condition

that F 1 = kF 2, where k is a constant.

3.2 Method of identifying potential Lax pairs

From Lax pairs of the type we consider here, the compatibility condition produces a

set of equations, each equation being due to one of the linearly independent spectral

terms that arises in some entry. Studies that have searched for integrable systems

by beginning with a Lax pair, whether an isospectral or isomonodromy Lax pair

or otherwise, typically assume some dependence of the Lax pair on the spectral

parameter, then solve the compatibility condition for the evolution equation [66, 63,

103, 46]. Most often a polynomial or rational dependence on the spectral variable is

used [104], but any type of explicit dependence could be investigated, for example

Weierstrass elliptic functions.

Example 1 Consider the following L and M matrices

L = a℘ 4b℘

c d℘ M =

α(℘2 + 1) β℘

γ℘ δ℘

where ℘ is the Weierstrass elliptic function in the spectral parameter n only, and

all other quantities are functions of both the lattice variables l and m. From the

39



compatibility condition, LM = M L, and noting that ℘2 = 14

℘3 − g2℘ − g3 where

gi = constant, we find the following equations in the 12 entry

12 ℘3 : aβ = bα + dβ

℘2 : bδ = 0

℘ : aβ = dβ − bα4g2

℘0 : aβ = dβ

(3.8)

where we have separated out the equations coming from different orders of the spectral

parameter.

Clearly, this choice of L and M does not yield an interesting evolution equation

through their compatibility, this example was instead chosen because it illustrates

an important point. Multiplying together two functions of the spectral parameter

can produce numerous orders that may or may not be proportional to other spectral

term products in the compatibility condition. From example 1, the spectral term

multiplying ℘2 did not ‘match up’ with other terms in the 12 entry at that order,

which brought about the equation bδ = 0, forcing some term to be zero. However,

the other spectral terms did turn out more meaningful equations, highlighting the

need to choose the dependence on the spectral parameter carefully so that none of

the resulting equations force any lattice terms to be zero.

The inclusion of a zero lattice term does not preclude the existence of an inter-

esting evolution equation. However, we are essentially classifying Lax pairs by the

number of terms in their entries and the class of Lax pairs presently under inspection

contains one separable term in each entry of their 2×

2 matrices. If any of those

terms were forced to be zero then the resulting Lax pair would actually come under

a different category in the present framework.

40



The most general form of a 2 × 2 Lax pair with exactly one separable term in

each entry of the L and M matrices is:

L = a A b B

c C d D

M =

α Λ β Ξ

γ Γ δ ∆

Where lower cases represent lattice terms and upper cases represent spectral terms.

The compatibility condition is LM = ML, of which we initially concentrate on the

12 entry.

aβAΞ + bδB∆ = bαBΛ + dβDΞ (3.9)

At this stage we are only concerned with the various linearly independent spectral

terms that appear. These will determine the set of equations that come out of the

compatibility condition, which are subsequently solved to find the corresponding

evolution equation. That being the case, there are four quantities to contend with

in this entry, AΞ, B∆, BΛ and DΞ. Any of these four products can lead to multiple,

linearly independent spectral terms, all of which must match up with at least one

other spectral term from one of the three remaining products in this entry. If

there exists some spectral term that does not match up with a spectral term from

another product, then the lattice term that multiplies it will have to be zero, which

is forbidden.

Let us label the terms in each product as follows AΞ =

i F AΞi, B∆ =

i F B∆i

,

etc. All the spectral terms that occur in the 12 entry of the compatibility condition

can be sorted into four groups according to the lattice terms that they multiply.

12 (aβ ) (bδ) | (bα) (dβ )

F AΞ1 F B∆1 | F BΛ1 F DΞ1

F AΞ2 F B∆2 | F BΛ2 F DΞ2

...... | ...

...

(3.10)

41



Where the line separating the four groups marks the position of the equals sign in

the associated lattice term equations. Organizing the terms from example 1 in this

way leads to

12 (aβ ) (bδ) | (bα) (dβ )

℘3 ℘2 | ℘3 ℘3

℘ | ℘ ℘

℘0 | ℘0

The present study will utilize the word ‘group’ in the general English sense, our

groups refer to collections of spectral terms that multiply the same lattice terms in

an entry of the compatibility condition. We have already seen that all terms in all

groups must be proportional to a term from at least one of the other three groups

in this entry. Conversely, where there are spectral terms that are proportional to

others from other groups, an equation relating the corresponding lattice terms will

thus be defined. Still with the above example, that Lax pair has a dependence on

the spectral parameter such that the groups multiplying aβ , bα and dβ all contain

the spectral term ℘3, for which the corresponding equation in the lattice terms is

aβ = bα + dβ .

At this stage, the number of possible sets of proportional spectral terms, and

therefore the number of possible combinations of equations yielded by the compati-

bility condition, is unmanageably large. We require further considerations to bring

the problem under control.

3.2.1 Links and equivalent equations at different orders

The word proportional is used in this chapter in the sense that two terms F 1 and

F 2 are proportional if F 1 = kF 2 for some finite constant k.

Definition 2 A link is a set of proportional spectral terms in the same matrix entry

of the compatibility condition. A set of two proportional terms is a single link, three

42



terms a double link, and four terms a triple link.

Naturally, the spectral terms that comprise a link must each reside in a different

group of spectral terms within an entry. If there are two or more spectral terms

that are proportional to one another within the same group, they are simply added

together to make one term. Since each group of spectral terms multiplies the same

lattice term, one may speak of either links between the groups of spectral terms or

links between lattice terms, with the same meaning. The above definition captures

the idea that the entries of the compatibility condition give rise to different lattice

term equations at different orders in the spectral term, without appealing to powers

of some basic function.

With the employment of proportional terms comes the possibility of constants

of proportionality, which we begin to deal with here.

Fact 1 If there exist two distinct single links between the same two groups of spectral

terms, then the corresponding constants of proportionality must be equal.

The proof of this fact is elementary: say that one single link is formed by allowing

F AΞ1 ∝ F DΞ1, where neither of these terms is proportional to any other spectral

term that arises in the 12 entry, and the other single link corresponds to F AΞ2 ∝F DΞ2, where again these terms link with no others. By including some constants

of proportionality, k1 and k2 respectively, we can write down the equations that

correspond to these two single links, those being aβ = k1dβ and aβ = k2dβ . Since

both equations must hold, and none of the lattice terms can be zero, it is clear that

we must have k1 = k2.

This simple fact proves to be rather important because it ensures that all links

between the same two groups can be bundled together. Further, all the spectral

terms, in some group, that correspond to those single links with one other group,

can be treated as a single spectral term. So, where F AΞ1 and F DΞ1 formed one single

43



link and F AΞ2 and F DΞ2 formed another between the same two groups, we can lump

together F AΞ1 + F AΞ2 = G1 and be sure that it links with F DΞ1 + F DΞ2 = kG1.

Still with the 12 entry, if there exist multiple double links between the same

three lattice terms, aβ , bα and dβ say, then the lattice term equations that result

from those links can be written

K

aβ

bα

dβ

= 0 (3.11)

Where K is a matrix of the constants of proportionality between the various spec-

tral terms, normalized so that each entry of the first column of K is unity. Each

row of K corresponds to a double link. If K is such that equation (3.11) is over-

determined or uniquely solvable, then any Lax pair possessing the corresponding

links is inconsistent or contains a zero lattice term. Therefore, we need not consider

more than two double links between the same three spectral terms, although there

may be multiplicity within those two double links. By the same argument we can

allow a maximum of three different triple links between the same four lattice terms

in an entry of the compatibility condition.

3.2.2 Link symbolism

The abundance of Lax pairs that need to be checked necessitates the introduction

of a shorthand, which will be based on their links. The off-diagonal entries both

contain four groups of spectral terms, each of which multiplies a single product of

lattice terms. The 12 entry contains the spectral term products AΞ, B∆, BΛ, and

DΞ associated with the lattice term products aβ , bδ, bα and dβ respectively. For

the shorthand, we always set out the spectral terms in the same way on the page

AΞ BΛ

B∆ DΞ

44



Each link can be represented by lines between the quantities that are proportional

to each other, and so we will use the symbols listed in table 3.1 to represent the

combinations of links in the 12 entry, where F AΞ is some term from the group of

spectral terms formed by taking the product AΞ, and other terms are similarly

labeled.

Symbol Links

F AΞ ∝ F DΞ, F B∆ ∝ F BΛ

F AΞ ∝ F B∆, F BΛ ∝ F DΞ

F AΞ ∝ F BΛ, F B∆ ∝ F DΞ

F AΞ ∝ F DΞ ∝ F B∆ ∝ F BΛF AΞ ∝ F B∆ ∝ F BΛi

, F BΛj∝ F DΞ

......

Table 3.1: Symbols used to represent the link combinations in the off-diagonal en-

tries. Note that F BΛi= kF BΛj

, k a constant

The 21 entry is similar to the 12 entry in that it possesses four distinct productsof spectral terms. The same symbols listed in table 3.1 are used again for the 21

entry, with clear meaning given that the spectral products are set out as follows

C Λ AΓ

DΓ C ∆

However, the diagonal entries are slightly different because each diagonal entry

contains a product that occurs twice: AΛ occurs twice in the 11 entry and D∆twice in 22. This automatically causes the associated lattice terms to be paired in

their respective entries and, as such, AΛ and D∆ need not be linked with another

spectral term to prevent a zero lattice term. This being the case, there are really

only three spectral term products to consider in both of the diagonal entries, one of

which need not be linked to the other two, and our symbols reflect that. Positioning

45



the spectral term products as follows

AΛBΓ

C Ξ

We symbolize the links as indicated in table 3.2, where the symbol ‘ ’ is used to

represent the repeated spectral term products.

Symbol Links

F AΛ ∝ F BΓ ∝ F C Ξ

F AΛ alone, F BΓ ∝ F C Ξ

F AΛi ∝ F BΓ and F AΛj ∝ F C Ξ separately......

Table 3.2: Symbols used to represent the link combinations in the diagonal entries

of the compatibility condition. Note that F AΛi= kF AΛj

, k a constant

3.2.3 Which Lax pairs need to be checked?

All link combinations that do not force any lattice terms to be zero are checked

systematically. The procedure for doing this runs as follows:

1. Begin with the 12 entry, assume that only single links exist there, and list all

single link combinations that produce different lattice term equations.

2. Construct the links found in the previous step by choosing proportional sets

of spectral terms in the appropriate groups.

3. Move to the 21 entry, noting the spectral term constructions from the previous

step, and identify all the viable link combinations in this entry.

4. Repeat the previous step in the diagonal entries.

46



After identifying all link combinations with single links in the 12 entry, we repeat

the entire process assuming that double and possible single links exist there, and

then repeat once more with triple, and possibly double and single links in the 12

entry. Finally, the corresponding set of lattice term equations for each Lax pair

must be analyzed to find the resulting evolution equation. This analysis needs to be

conducted in a manner that preserves the full freedom of the system, as described

in section 3.3. Note that this final step of the procedure is not conducted in the

present section, here we only identify the Lax pairs that need to be analyzed, we

leave the analysis itself to sections 3.3 and 3.4.

Single links in the 12 entry

The four groups of spectral terms in the 12 entry are

AΞ BΛ

B∆ DΞ

Each group must be linked to another so, using single links, the group AΞ must be

linked to at least one of the three other groups, BΛ, B∆, or DΞ. For arguments

sake, say that there exists a single link between AΞ and BΛ.

AΞ BΛ

B∆ DΞ

That leaves both groups B∆ and DΞ requiring links and, since we are only concerned

with single links at the moment, these two groups of spectral terms can be linked

to each other, or to one of AΞ or BΛ, in distinct, single links.

At this point, to make the number of Lax pairs that are to be scrutinized later

more manageable, we segregate a certain class of Lax pairs with single links. It can

be shown that if any group in either off-diagonal entry possesses single links between

it and two other groups, the resulting Lax pair is associated with a trivial evolution

equation (see Proposition 3.1 below).

47



Proposition 3.1 If there exist two (or more) single links between some lattice term

and two others in an off-diagonal entry, then the resulting evolution equation is

trivial.

Proof 3.1 The proof of Proposition 3.1 lies in checking all the possible link combi-

nations that meet the criterion. This is not written down here but can be done in a

way that is analogous to the analysis of the other link types.

Proposition 3.1 is used to exclude a class of Lax pairs from explicit analysis, we do

not take it any further here. There is nothing inherently special about this class

of Lax pair, it is only treated separately because it contains many cases that are

similar to others studied below, so including these would unnecessarily lengthen the

argument, see table 3.3 for examples of the types of links combinations excluded by

Proposition 3.1.

12AΞ BΛ

B∆ DΞ. . .

Table 3.3: Proposition 3.1 excludes Lax pairs that have two distinct single links

between some entry and two others in 12 entry

Hence, without considering the link combinations excluded by Proposition 3.1

and given the first link between AΞ and BΛ, the only other single link that needs

to be considered is between B∆ and DΞ. A similar argument holds when we choose

a single link between AΞ and B∆ or between AΞ and DΞ, therefore, table 3.4 liststhe only single link combinations in the 12 entry that require further analysis.

We shall proceed with the analysis under the assumption that single links be-

tween AΞ and BΛ, and between B∆ and DΞ, although the other combinations can

be dealt with in the same way. The specified links are constructed by choosing

48



12AΞ BΛ

B∆ DΞ

Table 3.4: Single link combinations in the 12 entry

spectral terms

A =1

Ξ(F 1 + . . .) (3.12a)

D =1

Ξ(F 2 + . . .) (3.12b)

Λ =1

B(F 1 + . . .) (3.12c)

∆ = 1B (F 2 + . . .) (3.12d)

where F i = F i(n) and F 1 = kF 2, k a constant. It is understood that while there is

room for other spectral terms in the expression A = 1Ξ

(F 1 + . . .), there cannot be a

term proportional to F 2/Ξ in A, as this would cause the single link corresponding

to F 2 to become a double link. In this way the desired links are constructed, plus

we have allowed for additional links should they be appropriate or required later.

Readers may note the omission of any constants of proportionality in the above,

however, the constants that could have been written at this point can all be absorbed

into the lattice terms that they multiply.

Turning our attention to the 21 entry of the compatibility condition, cαC Λ +

dγDΓ = aγAΓ + cδC ∆, the following spectral terms appear

21 (cα) (dγ ) | (aγ ) (cδ)

C B

F 1ΓΞ

F 2

|ΓΞ

F 1C B

F 2

... ... | ... ...

(3.13)

Since every spectral term must be proportional to another in the same entry,

C B

F 1 must be proportional to ΓΞ

F 1 or ΓΞ

F 2. Clearly, C B

F 1 cannot be proportional

to C B

F 2, nor can it link with some other term that we are yet to define, as this

would introduce single links between some lattice term and two others, thus leading

49



to a trivial evolution equation by Proposition 3.1. Moreover, we can exclude the

case with C B

F 1 ∝ ΓΞ

F 2 because this also requires the remaining spectral terms to be

proportional to one another, i.e. C B

F 2 ∝ ΓΞ

F 1. These two conditions on the spectral

terms imply that F 1 ∝ F 2, contradicting a previous assumption. Hence, the links

chosen in the 12 entry leave only one choice for the links in the 21 entry of the

compatibility condition: C B

F 1 ∝ ΓΞ

F 1 and C B

F 2 ∝ ΓΞ

F 2, which can be written more

succinctly as C Ξ ∝ BΓ

12 AΞ BΛ 21 C Λ AΓ

⇒

B∆ DΞ DΓ C ∆

The terms arising in the diagonal entries are listed below

11 |(aα − aα) F 21 /(BΞ) | (cβ − bγ ) BΓ

|

22 |(dδ − dδ) F 22 /(BΞ) | (bγ − cβ ) BΓ

|

(3.14)

Note that the terms in equation (3.14) are set out slightly differently to those in

table 3.2 because we have already determined that BΓ ∝ C Ξ.

The spectral terms in the diagonal entries, in this case, do not necessarily have

to link with others because they are multiplied by more than one lattice term, asindicated in equation (3.14) where the lattice terms appear in parentheses to the

left of the spectral terms they multiply. A lone spectral term in the diagonal entries,

given the links already constructed in the off-diagonal entries, will not bring about

a zero lattice term. Consequently, there exist three possible link combinations for

the diagonal entries: BΓ ∝ F 21 /(BΞ), BΓ ∝ F 22 /(BΞ) or BΓ is proportional to

50



neither F 22 /(BΞ) nor F 21 /(BΞ). All three choices lead to trivial evolution equations

and we shall continue the analysis under the assumption that BΓ ∝ F 21 /(BΞ). A

link combination has now been chosen in each of the entries of the compatibility

condition, these are shown in table 3.5

12

AΞ BΛ

B∆ DΞ

21

C Λ AΓ

DΓ C ∆

11

AΛ

BΓ

C Ξ

22

D∆

BΓ

C Ξ

Table 3.5: An example of the links that define a Lax pair

Gauge transformations can be used to remove the dependence on some of the

spectral terms and, as such, we expect some redundancy. The links constructedabove are achieved by setting the values of the spectral terms to

A = F 1, Λ = F 1

B = 1, Ξ = 1

C = F 21 , Γ = F 21

D = F 2, ∆ = F 2

where F 1 and F 2 are any functions of the spectral variable n, such that F 1= kF 2,

k = constant. Note that one of F 1 or F 2 may itself be a constant. No constants

of proportionality are required as these can be absorbed into lattice terms in this

case. The same links are reproduced by any suite of spectral terms that satisfies the

51



following conditions, brought about by the links described above

A = F 1/Ξ, Λ = F 1/B

C = F 21

/Ξ, Γ = F 21

/B

D = F 2/Ξ, ∆ = F 2/B

The resulting set of equations that are produced by the compatibility condition

for this Lax pair are written in (3.15), although they can be read from the links

given in table 3.5.

aα− aα = cβ − bγ

dδ − dδ = 0

bγ − cβ = 0

aβ = bα

bδ = dβ

cα = aγ

dγ = cδ

(3.15)

This rounds the description of the Lax pairs with only single links in the off-

diagonal entries, in practice one would continue by analyzing equations (3.15) tofind the associated evolution equation, which in this case is trivial (for more on

the analysis systems of equations leading to a trivial evolution equation see section

3.4.1). On the possibility of including extra spectral terms to augment the links

used here, see equation (3.12), we note that fewer terms allow greater freedom and

that any additional links could only lead to a more constrained system, one that

certainly could not sustain an interesting evolution equation considering that the

less constrained example here leads to a trivial result. Also, the alternative linksthat cause there to be one equation in the 22 entry and two equations in the 11 entry

of the compatibility condition, see after equation (3.14), lead to the same evolution

equations found here as the Lax pair is symmetric in that sense.

52



Double links in the 12 entry

Here link combinations that consist of double and possibly single links in the 12

entry are investigated. There are more possibilities in this class than when onlyconsidering single links, however the number is reduced by noting that some sets of

link combinations are equivalent from the perspective of the lattice term equations.

For example the following pair of link combinations in the 12 entry are clearly

equivalent when considered as proportionality statements.

12AΞ BΛ

B∆ DΞ↔

AΞ + k1B∆ = k2BΛ

B∆ = k3DΞ↔ AΞ = k4BΛ + k5DΞ

B∆ = k3DΞ(3.16)

Also, all link combinations that possess two double links in the 12 entry are

nearly equivalent, enough to consider them all together, what is meant by ‘nearly

equivalent’ is clarified in the next two sentences. The equivalence is because the

lattice term equations corresponding to any two double links in this entry can be

manipulated so that the they are the same as those from any other combination of

two double links. The difference that may arise comes from the diagonal entries,

where the particular pair of double links chosen in the 12 entry can affect the variety

of link combinations possible. However, the difference is not sufficient to alter the

overall outcome that these systems are overdetermined. The complete list of double

link combinations in the 12 entry that need to be analyzed is shown in table 3.6

12AΞ BΛ

B∆ DΞ

Table 3.6: All double link combinations in the 12 entry that require analysis

53



To exemplify the method of construction of Lax pairs with double links in the 12

entry, we choose link combination in table 3.6 where there is a double link between

AΞ, B∆ and BΛ and a single link between B∆ and DΞ. Spectral terms that

generate this choice of links are given in equation (3.17).

A = F 1/Ξ, Λ = F 1/B

∆ = (F 1 + kF 2)/B, D = F 2/Ξ(3.17)

where our usual nomenclature applies, i.e. F i = F i(n), k = constant.

Using the expressions found in the 12 entry, the terms that arise in the 21 entry

are F 1C/B, F 2Γ/Ξ, F 1Γ/Ξ and F 2C/B, where F 1C/B arises twice. It is convenient

to rearrange the terms found in the 21 entry into columns of terms that cannot belinked, this is done in equation (3.18).

F 1C/B F 1Γ/Ξ

F 2Γ/Ξ (F 1 + F 2)C/B⇒ F 1Γ/Ξ F 1C/B

F 2Γ/Ξ F 2C/B(3.18)

As in section 3.2.2, the ‘ ’ is used to symbolize a repeated spectral term product

that does not necessarily have to be linked. F 2Γ/Ξ must link with F 2C/B because

linking with F 1C/B would lead to the contradictory F 1 ∝ F 2, since that would

also necessitate a link between F 2C/B and F 1Γ/Ξ. That leaves two possibilities:

BΓ ∝ C Ξ, or we can split F 2C/B ∝ (F 1 + F 2)Γ/Ξ, noting that the multiplicity of

the term F 1C/B means that it need not link to another in this entry. The second

possibility is neglected, though, as it gives rise to a zero lattice term in one of the

diagonal entries to be considered below. Thus, by a process of elimination, the links

selected in the 12 entry leave only one choice for the 21 entry, which is shown in

equation (3.19)

12 AΞ BΛ 21 C Λ AΓ

⇒B∆ DΞ DΓ C ∆

(3.19)

Move now to the 11 entry where the relevant spectral term products are AΛ ∝F 21 /(BΞ), and BΓ ∝ C Ξ, see equation (3.20). We choose to link these two spectral

54



terms to get one equation in the 11 entry and find that there are necessarily two

equations in the 22 entry. Note that the opposite case where there are two equations

in the 11 entry and one in the 22 entry can also exist by splitting up BΓ into two

terms, however this will lead to the same evolution equation.

11 |(aα − aα) F 21 /(BΞ) | (cβ − bγ ) BΓ

|

22 F 1F 2/(BΞ) |

(ˆdδ − d

¯δ) | (bγ − cβ ) BΓ

F 22 /(BΞ) |

(3.20)

Using our symbolism, the links that define this Lax pair are shown in table 3.7.

12

AΞ BΛ

B∆ DΞ

21

C Λ AΓ

DΓ C ∆

11

AΛ

BΓ

C Ξ

22

D∆

BΓ

C Ξ

Table 3.7: Links for a Lax pair with double links in the off-diagonal entries

The spectral term relations that must be satisfied to bring about the links in

55



table 3.7 are given in (3.21).

A = F 1, Λ = F 1

B = 1, Ξ = 1

C = F 21 , Γ = F 21

D = F 2, ∆ = F 1 + kF 2

(3.21)

where F 1 and F 2 are any functions of the spectral variable n, such that F 1 = kF 2,

k = constant. Notice that a constant appears in the expression for ∆ in equation

(3.21), since one of the constants of proportionality in this expression cannot be

absorbed into the multiplying lattice term, δ.

The lattice term equations that arise via the compatibility condition from this

Lax pair are shown in equation (3.22).

aα− aα = cβ − bγ

dδ − dδ = 0

bγ − cβ = 0

aβ + bδ = bα

bδ = kdβ

cα = dγ + cδ

aγ = kcδ

(3.22)

We thus conclude the description of the formation of Lax pairs with at least one

double link in the 12 entry. Naturally there are other Lax pairs of this type but they

are formed in a similar manner to that described here. A description of the analysis

of the systems of equations produced by the compatibility of these Lax pairs is leftuntil sections 3.3 and 3.4, although no Lax pairs of this type lead to unconstrained,

nonlinear evolution equations.

56



A triple link in the 12 entry

Lastly, link combinations including triple, and possibly double and single links in

the 12 entry must be considered. It is not difficult to see that the only possibilitythat needs to be investigated is that with one triple link between all four lattice

terms in the 12 entry, any additional links in this entry constrain the problem too

heavily and lead to trivial evolution equations only.

A triple link in the 12 entry of the compatibility condition can be formed by

setting the spectral terms to those in equation (3.23) below.

A = F 1/Ξ, Λ = F 1/B

∆ = F 1/B, D = F 1/Ξ⇒

12 AΞ BΛ

B∆ DΞ

(3.23)

Given the values in (3.23), the resulting spectral terms that appear in the 21

entry are as shown in equation (3.24).

21 C Λ AΓ

DΓ C ∆ ⇒F 1C/B F 1Γ/Ξ

F 1Γ/Ξ F 1C/B

(3.24)

We therefore have two possibilities in the 21 entry depending on whether or not

BΓ ∝ C Ξ. If BΓ is proportional C Ξ then there is another triple link in the 21

entry, otherwise there is a pair of single links. The latter case yields the links given

in item 4 of table 3.8 and leads to a trivial evolution equation, while the former

yields Lax pairs that include one for the LMKdV2 system.

We continue the analysis here assuming BΓ ∝ C Ξ, in this case the spectral terms

57



in the diagonal entries are shown in equation (3.25).

11 |(aα

−aα) F 2

1/(BΞ)

|(cβ

−bγ ) BΓ

|

22 |(dδ − dδ) F 21 /(BΞ) | (bγ − cβ ) BΓ

|

(3.25)

Again, two possibilities present themselves, this time depending on whether F 21 /(BΞ) ∝BΓ. The case where the proportionality does not hold provides a Lax pair for the

LMKdV2 system, which is discussed in section 3.3. When F 21 /(BΞ) ∝ BΓ does hold,

a unique situation unfolds where each entry of the compatibility condition contains

only one equation. This special case is discussed in section 3.4.4.

3.2.4 List of link combinations

We are now in a position to tabulate the results. Table 3.8 contains a representative

selection of all possible link combinations for 2×2 Lax pairs with a single, separable

term in each entry of the L and M matrices. There are still other link combinations

that were analyzed but do not appear in table 3.8 because they are equivalent to

a combination that does appear, or because it is clear that the corresponding Lax

pair cannot yield an interesting evolution equation since a similar, less constrained

combination of links is listed as trivial or over-determined.

Items 1 and 2 in table 3.8 are analyzed thoroughly in section 3.3. The other repre-

sentative link combinations are dealt with in section 3.4.

58



12

AΞ BΛ

B∆ DΞ

21

C Λ AΓ

DΓ C ∆

11

AΛBΓ

C Ξ

22

D∆BΓ

C Ξ

Evolution Eqn

1 LSG2 (3.2)

2 LMKdV2(3.4)

3 Trivial

4 Trivial

5 Zero term

6 Trivial

7 Underdetermined

8 Trivial

9 Trivial

10 Overdetermined

11 Trivial

12 Overdetermined

13 Overdetermined

14 Overdetermined15 Special case

Table 3.8: List of link combinations used to construct possible Lax pairs

3.3 Derivation of higher order LSG and LMKdV

systems

While fourteen potentially viable types of Lax pairs were identified in section 3.2,

only two types lead to non-trivial, well determined evolution equations. The two

systems thus found are LMKdV2 and LSG2 and this section describes the derivation

of these systems from the general form of their Lax pairs. It is important to note that

no freedom in the lattice terms is lost through this process, all values that the lattice

59



terms take are dictated by the sets of equations effectuated by the compatibility

condition. As such we conclude that the systems so derived (or equivalent systems)

are the most general ones that can be associated with their Lax pairs.

In fact, such calculations have been performed many times before and yet neither

LSG2 or LMKdV2 appear to have been published, despite coming from Lax pairs

with simple forms that have certainly already been considered previously [102, 105].

Hence, it is necessary to outline the method used to derive LSG2 and LMKdV2 in

detail.

3.3.1 LSG2

For LSG2 the Lax pair used as a starting point is

L =

F 1a F 2b

F 2c F 1d

(3.26a)

M =

F 2α F 1β

F 1γ F 2δ

(3.26b)

Where a, b, c, d, α, β , γ and δ are all functions of both the lattice variables l and

m, referred to as lattice terms, and F 1 and F 2 depend on the spectral variable only,

we refer to these terms as spectral terms.

It is possible to remove some of the lattice terms using a gauge transformation.

While this would reduce the complexity of the system, it is not clear at this point

which of the lattice terms would best be removed. Experience shows that natural

transformations present themselves in the course of solving the compatibility condi-

tion and so we shall wait until later to remove some lattice terms, keeping in mind

that we expect some freedom to disappear from each of the Lax matrices L and M .

The compatibility condition for (3.26), LM = ML, leads to the following system

60



of difference equations in the lattice terms

aα + bγ = aα + cβ (3.27a)

dδ + cβ = dδ + bγ (3.27b)

aβ = dβ (3.27c)

bδ = bα (3.27d)

cα = cδ (3.27e)

dγ = aγ (3.27f)

Some of equations (3.27) are linear and some nonlinear. The linear equations

can be solved easily when in the form

k1φ − k2φ = k3ψ − k4ψ (3.28)

where φ and ψ are lattice terms and ki constants. Equation (3.28) implies

ψ = k1v − k2v + µ(k4k3

)l

φ = k3v − k4v + λ(

k2

k1 )

m

where we have introduced the new lattice term v = v(l, m), and λ = λ(l) and

µ = µ(m) are constants of integration. The same fact also applies in a multiplicative

sense, in particular φφ

=ψ

ψ⇒ φ = λ

v

v, ψ = µ

v

v

We now proceed to solve the system (3.27). Equations (3.27c) to (3.27f) are

linear and may solved in pairs as follows. Multiply equations (3.27c) by (3.27f) to

find ad/(ad) = β γ/(βγ ), which is integrated for

a = λ20v5/(v5d)

β = µ5v5/(v5γ )

61



where we have introduced λ0 = λ0(l), µ5 = µ5(m) and v5 = v5(l, m). Now use

equation (3.27c) again to find v5γ/(v5γ ) = dλ0

dλ0

and integrate for

d = λ0ρv2/v2

γ = µ2v2v2/v5

where ρ = λ3(l)(−1)m

. Substitute these values back into the expressions for a and β ,

and replace v5/v2 → v1, and µ5/µ2 → µ1 resulting in

a =λ0

ρ

v1v1

(3.29a)

d = λ0ρv2v2

(3.29b)

β = µ1v1v2

(3.29c)

γ = µ2v2v1

(3.29d)

Perform similar calculations on equations (3.27d) and (3.27e) to find

b = λ1v3v4

(3.30a)

c = λ2

v4

v3 (3.30b)

α =µ0

σ

v3v3

(3.30c)

δ = µ0σv4v4

(3.30d)

Where σ = µ3(m)(−1)l

.

When equations (3.29) and (3.30) are substituted into (3.27a) and (3.27b) we

find the following equations respectively

λ0ρµ0

σ

ˆv1v3v1v3

+ λ1µ1v2v3v1v4

=λ0

ρµ0σ

v1ˆv3v1v3

+ λ2µ2

ˆv1v4v2v3

(3.31a)

λ0

ρµ0σ

ˆv2v4v2v4

+ λ2µ2

ˆv4v1v3v2

= λ0ρµ0

σ

v2ˆv4v2v4

+ λ1µ1

ˆv2v3v1v4

(3.31b)

Multiplying (3.31a) by v1/ˆv1 and (3.31b) by v4/ˆv4 indicates that certain variables

always appear in combination. As such, we set v1, v2 ≡ 1 without loss of generality,

62



and rename v3 = x and v4 = y. This is the manifest reduction in freedom that was

expected from the perspective of gauge transformations. The parameter functions

similarly appear in ratios, hence we set λ0 ≡ 1 and µ0 ≡ 1 without loss of generality.

Making the substitutions we arrive at a pair of nonlinear partial difference equations

in x and y, with arbitrary non-autonomous terms λi(l) and µi(m), which together

form LSG2, the evolution equation associated with the Lax pair.

ρ

σ

x

x+ λ1µ1 ˆxy =

σ

ρ

ˆx

x+

λ2µ2

xy(3.32a)

σ

ρ

ˆy

y+

λ2µ2

xy=

ρ

σ

y

y+ λ1µ1xˆy (3.32b)

This pair of equations can be thought of as a higher order lattice sine-Gordon system

because the lattice sine-Gordon equation (LSG) is retrieved by setting y = x in either

expression

LSG : ˆxx

σ

ρ− λ1µ1xx

=

ρ

σxx − λ2µ2

The Lax pair (3.6) for LSG2 is obtained by substituting the calculated values of the

lattice terms back into (3.26).

3.3.2 LMKdV2

The general form of the Lax pair for LMKdV2 is similar to that for LSG2, the only

difference being that here both matrices exhibit the same dependence on the spectral

variable, while the dependence was antisymmetric in the previous case.

L = F 1a F 2b

F 2c F 1d (3.33)

M =

F 1α F 2β

F 2γ F 1δ

(3.34)

As above, a, b, c, d, α, β , γ and δ are all functions of the both the lattice variables

l and m, F 1 and F 2 depend on the spectral variable only.

63



The compatibility condition leads to six equations coming from the different

orders of the spectral variable in each entry.

aα = aα (3.35a)

bγ = cβ (3.35b)

dδ = dδ (3.35c)

cβ = bγ (3.35d)

aβ + bδ = bα + dβ (3.35e)

cα + dγ = aγ + cδ (3.35f)

Equations (3.35a) and (3.35c) are integrated immediately, while (3.35b) and (3.35d)are multiplied together and dealt with by a similar method to that used for the

LSG2 system of section 3.3.1 leading to the following results

a = λ1v1/v1 (3.36a)

b = λ0ρv3/v4 (3.36b)

c = λ0v4/(ρv3) (3.36c)

d = λ2v2/v2 (3.36d)

α = µ1v1/v1 (3.36e)

β = µ0v3/(σv4) (3.36f)

γ = µ0σv4/v3 (3.36g)

δ = µ2v2/v2 (3.36h)

Where vi = vi(l, m), λi = λi(l), µi = µi(m), ρ = λ(−1)m3 and σ = µ

(−1)l3 . Substituting

these values into the two remaining equations, (3.35e) and (3.35f), shows that terms

consistently appear in ratios, as they did with the LSG2 system. We choose to set

v2 ≡ v3 ≡ 1, λ0 ≡ µ0 ≡ 1, v1 = x and v4 = y and arrive at LMKdV2:

λ1

σ

ˆx

x+

µ2

ρ

y

y= λ2σ

y

y+ ρµ1

ˆx

x(3.37)

ρµ1xˆy + λ2σxy =µ2

ρxy +

λ1

σxy (3.38)

64



3.4 How most link combinations lead to bad evo-

lution equations

Here we explain how most of the potentially viable Lax pairs fail to produce inter-

esting evolution equations. The topic is split into four parts dealing with Lax pairs

that lead to trivial, over-determined, under-determined evolution equations and the

special case of item 15 in table 3.8.

3.4.1 Lax pairs that yield only trivial evolution equations

This section is pertinent to items 3, 4, 6, 8, 9 and 11 in table 3.8, those Lax pairs

whose compatibility conditions can be solved to a point where only linear equations

remain, or the equations can be reduced to first order equation in one lattice direction

only.

The simplest route to triviality is to have a Lax pair with a set of equations that

are all linear, as per item 3 in table 3.8. The compatibility condition leads to a set

of eight linear equations in the eight initial lattice terms.

aα = aα, bγ = cβ

dδ = dδ, cβ = nγ

aβ = dβ, bδ = bα

dγ = aγ, cα = cδ

(3.39)

Equations (3.39) can be solved easily using techniques described in section 3.3,

with the result of a simple, linear evolution equation. However, it is not necessary to

conduct such analysis on this system since all equations (3.39) are linear and they

can not be expected to produce a nonlinear evolution equation. For this reason,

other examples of link combinations that produce only linear equations have been

omitted from table 3.8.

65



Item number 8 from table 3.8 is an example of a Lax pair that leads to a trivial

evolution equation in a more complex way. The equations that come out of its

compatibility condition are

aα = aα (3.40a)

bγ = cβ (3.40b)

dδ + cβ = dδ + bγ (3.40c)

aβ + bδ = bα (3.40d)

kbα = −dβ (3.40e)

cα = aγ + cδ (3.40f)

kcα = −dγ (3.40g)

Where k is a constant. This system of equations is solved as follows: integrate

(3.40a) for a = λv/v, α = µv/v, introducing λ = λ(l), µ = µ(m) and v = v(l, m).

Multiply equations (3.40b) and (3.40g), then divide the product by (3.40e) and use

the values calculated for a and α to find

bcv

bcv =

dˆv

dv . (3.41)

Equation (3.41) is integrated for b, which is substituted into equations (3.40e) and

(3.40b) yielding the following values

b = λ2dv

cv

β = −kµλ2

ˆv

cv

γ =−

kµcv

dv

There are now three equations left to solve, (3.40c), (3.40d) and (3.40f), where it is

found that c and v always appear as a product which suggests we introduce u = cv.

Equation (3.40c) is then used to write

δ = µu

u+ kµλ

ˆu

u ˆd

66



Finally, we introduce x = ud/u and achieve the final two equations in x

λ2(x− x) = k(λ2λ − λλ2) (3.42a)

x−

x + k(λ−

λ2) = kx

ˆx(µλ

−λ2) (3.42b)

Equations (3.42) is an overdetermined set of two equations in the one variable, x,

which cannot hope to yield an interesting evolution equation. This is because (3.42a)

can be used to remove the dependence of x on m (m is the independent variable of

the ‘ˆ’ direction), leaving x with a first order dependence on l in (3.42b) at best.

3.4.2 Overdetermined Lax pairs

Here items 10 and 12 through 14 in table 3.8 are dealt with. Such Lax pairs have

compatibility conditions that boil down to more equations than there are free lattice

terms. These are not necessarily trivial, as the solution to one equation may solve

another as well, that is it may be possible to make one or more equations redundant.

Some systems of this type, where any hope of supporting an interesting evolution

equation has been quashed, have already been considered in section 3.4.1. The

remaining over-determined systems are considered here, however we do not attempt

to resolve the issue of whether these systems support interesting evolution equations,

we simply list them as being overdetermined.

The most interesting instances of this type of system arise from Lax pairs with

two double links in the off-diagonal entries, those represented by item 14 from table

3.8. All such systems, with minimum constraint on the lattice terms in the diagonal

entries of the compatibility condition, lead to the same evolution equations. An

example of a Lax pair with two double links in the off-diagonal entries is

L =

F 1a b

(F 21 + k4F 1F 2)c F 2d

(3.43a)

M =

(F 1 + k2F 2)α β

(F 21 + k3F 1F 2)γ (F 1 + k1F 2)δ

(3.43b)

67



where lower case letters except ki are lattice terms, ki are constants of proportion-

ality and F i are spectral terms with F 1 not proportional to F 2. This particular

Lax pair is especially interesting because the solution of its compatibility condition

involves integrating both additive and multiplicative linear difference equations, as

described near equation (3.28) in section 3.3. The final evolution equations achieved

are relatively complicated and nonlinear, however there are two equations for one

variable and it remains to be seen whether they can be reconciled. The following

outlines how to solve the compatibility condition.

In the compatibility condition,

LM = ML, the 11 entry dictates k2 = k3 = k4.

Redefine F 2 → k2F 2 to remove k2 from everywhere except the 22 entry of L wherewe set k0 = 1/K 2 and thus arrive at the following equations from the various orders

of the compatibility condition

aα + bγ = aα + cβ (3.44a)

cβ = bγ (3.44b)

dδ = dδ (3.44c)

aβ +ˆbδ = bα (3.44d)

k1bδ = bα + k0dβ (3.44e)

cα = aγ + cδ (3.44f)

cα + k0dγ = k1cδ (3.44g)

Integrate (3.44c) for d = λv/v, δ = µv/v, introducing v = v(l, m), λ = λ(l) and

µ = µ(m), and use equation (3.44b) to see that β = bγ/c. Now define s = c/v,

t = γ/v and u = bv so that

β = bt/s (3.45a)

(3.44g) ⇒ α =1

s(k1µs− k0λt) (3.45b)

(3.44f ) ⇒ a =1

t((k1 − 1)µs − k0λt) (3.45c)

68



Substituting these (3.45a) values into equation (3.44e) and rearranging leads to

k1

us

λλ

− k1

us

λλ

= k0

ut

λµ

− k0

ut

λµ

(3.46)

which is an additive linear difference equation that can be integrated to find

us

λλ= w − w + λ2 (3.47a)

ut

λµ= w − w + µ2 (3.47b)

The two equations that remain, equations (3.44a) and (3.44d) respectively, are

written in terms of s, t and u

ˆs

ˆt((k − 1)µ − k0λ

t

s)(k1µs

t − k0λ) + uˆs = ((k − 1)µs

t − k0λ)(k1µ− k0λt

s) + u¯ts

t (3.48a)

ˆs

ˆt((k − 1)µ− k0λ

t

s) + µ

uˆs

ut= k1µ

s

t− k0λ (3.48b)

Make a further change of variables x = ut, y = us, to completely remove u from

equations (3.48) and (3.47). Equation (3.48b) becomes

µˆy

x+

ˆy

ˆx((k1 − 1)µ− k0λ

x

y) = k1µ

y

x− k0λ (3.49)

while equation (3.48b) becomesˆy

ˆx((k1−1)µ−k0λ

x

y)(k1µ

y

x−k0λ)+ ˆy = ((k1−1)µ

y

x−k0λ)(k1µ−k0λ

x

y)+ ¯x

y

x(3.50)

where equation (3.47) shows that x and y can both be written in terms of w according

to

x =λµ

k0(w − w + µ2), (3.51a)

y =λλ

k1(w

−w

−λ2). (3.51b)

Hence, there are two complicated, nonlinear equations for the one variable w, (3.49)

and (3.50) and the system is therefore overdetermined. However, these two equations

may or may not be reconcilable, one way that they might be reconcilable is if one

equation was shown to be a compatible similarity constraint for the other, in the

sense of [82, 78].

69



3.4.3 Underdetermined Lax pairs

There are Lax pairs whose compatibility condition can be solved completely, while

still leaving at least one lattice term free. In these cases there is no genuine evolutionequation, although the freedom inherent in the system could to cause it to appear

as though there was. In fact, any evolution equation, trivial, integrable or even

chaotic, could be falsely represented by such a Lax pair. One such case is item 7 on

table 3.8 that has as its compatibility condition

aα + bγ = aα + cβ (3.52a)

ˆdδ + cβ = d

¯δ + bγ (3.52b)

aβ = −bδ (3.52c)

bα = −dβ (3.52d)

cα = −dγ (3.52e)

aγ = −cδ (3.52f)

Here is a roadmap to the solution: multiply equation (3.52c) by (3.52f) and divide

by (3.52d) and (3.52e) to find an expression that can be integrated for

d = λ1vcb/(va)

γ = µ1vαδ/(vβ )

Substituting these values for d and γ into the ratio of equations (3.52d) and

(3.52f) shows that v = λ1µ1ˆv, which indicates that v must be separable into a

product such as v = λ2(l)µ2(m), where λ1 = λ2/¯λ2 and µ1 = µ2/µ2. With the above

values included and no further integration required, equation (3.52c) is used to find

70



δ while (3.52d) offers c. In summary:

d = −bα/β (3.53a)

γ = ±aα/b (3.53b)

δ = −aβ/b (3.53c)

c = ±aα/β (3.53d)

The key feature is that, with all negative quantities in (3.53), equations (3.52a)

and (3.52b) are automatically satisfied, furnishing us with no more constraints on

the remaining lattice terms a, b, α and β . Therefore, equations (3.53) are the

only conditions that must be met in order to satisfy consistency, but these are

not uniquely determined and there is freedom enough to write any equation at all

into this set. The undeniable conclusion is that any Lax pair of this type is false

as one would expect that the information to be gleaned about the solution to any

evolution equation associated with this Lax pair must be as vague as equation (3.53).

Note further that the level of freedom left in the system is exactly that which can

be removed by gauge transformations, so, in essence, this system’s compatibility

is simply determined by the values of the parameter functions and no evolution

equation exists.

When c and γ in equations (3.53) are positive the situation is slightly more

complicated because equations (3.52a) and (3.52b) are not automatically satisfied.

However, the qualitative outcome is the same and the system remains underdeter-

mined.

3.4.4 A special case

Item 15 from table 3.8 is a special case that sees all lattice terms linked in all four en-

tries of the compatibility condition. Thus, the compatibility condition supplies only

nonlinear equations, none of which can be explicitly integrated, and so no parameter

71



functions present themselves as constants of integration. This is an unusual situa-

tion but it is not necessary to solve the system because of the following arguments

regarding the spectral dependence.

To form the links that define this Lax pair we require the following conditions,

or an equivalent set, on the spectral terms.

A = BΓ/Λ Ξ = Λ2/Γ

C = BΓ2/Λ2 ∆ = Λ

D = BΓ/Λ

(3.54)

Equation (3.55), below, displays a Lax pair of general form that possesses the links

in question

L = BF 1

a b/F 1

cF 1 d

, (3.55a)

M = Λ

α β/F 1

γF 1 δ

, (3.55b)

where F 1(n) = Γ/Λ. The prefactors in equation (3.55) are irrelevant because they

cancel in the compatibility condition. This leaves the Lax pair with the same de-

pendence on just one spectral term in both the L and M matrices. As such, a gauge

transformation can be used to completely remove the dependence on the spectral

variable from the linear problem, implying that equation (3.55) is actually not a Lax

pair at all.

3.5 Discussion

All 2 × 2 Lax pairs, with one separable term in the four entries of each matrix,

have been considered through the various combinations of terms possible in their

compatibility conditions. It has been shown that the only non-trivial evolution

equations that can be supported are the higher order generalizations of the LMKdV

72



and LSG equations, with the possible exception of over-determined systems that

may yet be consistent, see section 3.4.2.

There is an important question in where the present work sits in relation to

studies in multidimensional consistency, or consistency around a cube (CAC), that

provide a method of searching for integrable partial difference equations where a Lax

pair can be derived as a bi-product of the procedure [38, 37]. The present work adds

to multidimensional consistency studies by removing the restriction that the Lax

matrices must be symmetric, a consequence of using the same Q equation in all three

directions of the cube [40], and also leads to equations with more non-autonomous

terms than multidimensional consistency studies have so far allowed. CAC studiesdo lead to non-autonomous equations, by allowing the parameters belonging to each

side of a face to vary only with one lattice variable, but they have not used any

non-autonomous terms that are arbitrary in one lattice variable and have a specific

dependence on another, like ρ and σ in equation (3.37). Non-autonomy is vital to

reductions of the type used in [86] and [2], that lead to nonlinear ordinary difference

equations and Lax pairs for them. In addition, while it has been shown that (CAC)

ensures the existence of a Lax pair, the converse is not necessarily true, which initself indicates a need for the present completeness study.

Future studies using the techniques explained here should investigate Lax pairs

with non-separable terms or with more terms in each matrix entry, and possibly aim

for a general algorithm for dealing with an arbitrary number of terms in each entry

of the Lax matrices. Other types, such as differential difference Lax pairs, should

be considered, as should purely continuous Lax pairs where there is already a vast

body of knowledge with which to compare results. Further, this technique is easily

adaptable to isomonodromy Lax pairs.

There is some conflict about whether the parameters of the Q4 equation, in the

ABS scheme, must lie on elliptic curves or not [36, 106, 107]. The Lax pair for Q4,

found via multidimensional consistency, is for a version of the equation where the

73



autonomous parameters are restricted to elliptic curves [38]. An exploration into

the links present in that Lax pair might resolve the issue by providing a Lax pair

for Q4 with non-autonomous, and possibly free, parameters.

74



Chapter 4

Reductions and Lax pairs for the

reduced equations

4.1 Introduction

Nonlinear evolution equations occur frequently in physical modeling and applied

mathematics. Nonlinear integrable lattices provide a natural discrete extension of

classically integrable systems. More recently, there has been great interest in non-

linear ordinary difference equations. We consider reductions from lattice equations

to ordinary difference equations which constitute a natural link between the two

classes of equations. Our main perspective will lie in the construction of Lax pairs

for difference equations.

Most studies [108, 78, 82, 109, 110, 111] of reductions of lattice equations focus on

equations in which all parameters are independent of lattice variables. For example,

the lattice modified Korteweg-de Vries equation [78]

LMKdV : xl+1,m+1 = xl,m

p xl+1,m − q xl,m+1

p xl,m+1 − q xl+1,m

contains lattice parameters p, q which are considered to be independent of the

75



lattice variables l, m. In [86], a new type of reduction from the lattice equations

to ordinary difference equations was introduced by starting with non-autonomous

lattice equations. In this approach, the lattice parameters p, q were considered

to be functions of l, m, under the condition that the lattice equation satisfied the

singularity confinement property. Note also that higher order versions of the non-

autonomous LMKdV and lattice sine-Gordon (LSG), with similar constraints on the

non-autonomous terms, were found in chapter 3 by an entirely different approach.

In [86], such non-autonomous forms of well-known lattice equations were shown to

reduce to q-discrete Painleve equations, including qPII, qPIII and qPV.

The q-discrete Painleve equations are of fundamental interest in the theory of integrable systems and random matrix theory. We note that the full generic form

of qPIII was first obtained in [33]. Its natural generalization is a q-discrete sixth

Painleve equation (qPVI) first obtained by [112]. The integrability of such equations

lies in the fact that they can be solved through an associated linear problem called

a Lax pair. For qPIII the Lax pair was obtained by [47], with a notable feature that

the linear problem is a matrix problem involving matrices of size 4×4. On the other

hand, the Lax pairs of lattice equations, such as the LMKdV [108, 78], and manydiscrete Painleve equations, such as qPVI are typically matrix problems of size 2×2.

In [113], a 2 × 2 Lax pair was given for a special case of qPIII .

In this chapter, we present two types of results. First, we show that an extension

of the reduction method given by [86] is possible and, by using the extension, deduce

a sequence of discrete Painleve equations as reductions of lattice equations. Second,

we give a Lax pair of the non-autonomous LMKdV and show that it gives rise to

2 × 2 matrix Lax pairs under the reductions to q-Painleve equations. In obtaining

the latter, a key observation was needed that arises from the multi-dimensional

embedding of lattice equations in a self-consistent way in three directions. The

resulting theory [89, 38, 37, 40] is often referred to as “consistency around a cube”.

The chapter is organized as follows. In section 4.2, we recall the Lax pair

76



of LMKdV and generalize it to provide a non-autonomous Lax pair for the non-

autonomous version of LMKdV. We also show that this Lax pair and the gener-

alized LMKdV form a multi-dimensional system that satisfies the self-consistency

property. In section 4.3, we consider the reductions of the non-autonomous LMKdV

to ordinary difference equations and provide extensions of previously considered re-

ductions. In section 4.4, we show that 2 × 2 Lax pairs for the reductions can be

found by applying the idea of self-consistency and reductions to the Lax pair of the

LMKdV. We end the chapter with a conclusion where we also point out some open

problems.

4.2 Lax Pair and Self-Consistency of the Non-

autonomous LMKdV

While a linear problem, or Lax pair, associated with the LMKdV has been known

for a long time [108, 78], it appears that linear problems associated with the non-

autonomous version of the LMKdV have not been written down before. We provide

an explicit Lax pair for the non-autonomous version of the LMKdV in the first

subsection below.

Furthermore, while the theory of multi-dimensional extensions of lattice equa-

tions has been explored fairly widely, the theory has not been applied explicitly

to non-autonomous lattice equations. We provide such an application to the non-

autonomous LMKdV in the second sub-section below.

77



4.2.1 Lax Pair of the Non-autonomous LMKdV

A Lax Pair for the LMKdV is a linear problem of the form

θ(l + 1, m) = L(l, m)θ(l, m),

θ(l, m + 1) = M (l, m)θ(l, m).(4.1)

whose compatibility condition, namely, L(l, m + 1) M (l, m) = M (l + 1, m) L(l, m),

is the LMKdV.

Hereafter we adopt the notation θ = θ(l + 1, m) and θ = θ(l, m + 1). (We

have used l in place of the more traditional n here because it is notationally more

appropriate that the L matrix should create a shift in l and M in m. Later, in §3,

we will see that a third Lax matrix, N arises whose associated shifts will be in n.)

Now set

L =

x/x −λ/(νx)

−λx/ν 1

, (4.2a)

M =

x/x −µ/(νx)

−µx/ν 1

. (4.2b)

where ν is a spectral variable, µ is a function of m alone, and λ is a function of l

alone.

Compatibility occurs when LM = ML. In this equation, it is straightforward

to check that the diagonal entries yield identities and the off-diagonal entries each

contain the lattice mKdV equation in the following way. The top-right entry yields

µˆx

xx +λ

x =λˆx

xx +µ

x ⇒ ˆx(µx − λx) = x(µx− λx).

Similarly, the bottom left entry yields the same equation. Thus we arrive at the

following form of the LMKdV equation

ˆx = xx − rx

x − rx(4.3)

78



where we have introduced

r(l, m) =µ(m)

λ(l). (4.4)

This form of the LMKdV equation is identical to the one used in [86], except for

a factor of (-1) which is inconsequential. Indeed, we can achieve the equation used

there exactly if we premultiply each of L and M by

1 0

0 −1

. We use the slightly

different form here simply because it allows for a more symmetric Lax Pair. In [86]

it is noted that r must separate as in (4.4) because it has to satisfy

ˆrr = rr (4.5)

for the singularity confinement property to be satisfied. We note that integrability

conditions for lattice equations have also been studied recently by [102].

4.2.2 Consistency Around a Cube

In this subsection, we show that the Lax Pair, (5.32), given in the previous subsec-

tion, is multi-dimensionally consistent with the lattice equation LMKdV.

In this point of view, the lattice variables l, m provide a two-dimensional slice

of a three-dimensional space in which the third direction, coordinatized by n say,

can be thought of as providing the spectral direction for the Lax pair. The shifts

l → l + 1, m → m + 1, n → n + 1 describe a fundamental cube in this multi-

dimensional space. The term “consistent around a cube” arises from the fact that

the iteration of the map on any face of the fundamental cube provides a corner value

that is the same as that provided by iteration on an intersecting face.

Define x = x(l,m,n + 1), such that x = u/t, where u and t are the components

of the eigenfunction θ(l,m,n), i.e.,

θ =

t

u

79



where θ satisfies the linear system (5.1).

Since θ = Lθ,

¯x = u/t

=u− λxt/ν

xt/x− λu/(ν x)

¯x = xx − ρx

x − ρx(4.6)

where we have allowed ν to depend on n and replaced λ/ν by ρ(l, n). Since M takes

on the same form as L, we can clearly find an equivalent expression in the m and n

directions. And, because (4.6) is the LMKdV equation again, we conclude that the

Lax Pair is multi-dimensionally consistent with the LMKdV equation.

Essentially we have done the reverse of the usual operation, ordinarily one begins

with a system that is consistent around a cube and then constructs its Lax Pair (see

[40] or [89, 38]). However, here we began with the Lax Pair and showed that it is

multi-dimensionally consistent with the LMKdV equation.

4.3 Reductions to Ordinary Difference Equations

In this section, we consider reductions from the partial difference equation (4.3) to

a sequence of ordinary difference equations. These include qPII, a three-parameter

version of qPIII , a special case of qPV, and, moreover, some higher-order difference

equations. We present the results in a series of subsections.

Let x = f (x) where x represents x and its iterates, x, ¯x... Thus, (4.3) becomes

f (x) = xx− rf (x)

f (x) − rx. (4.7)

For f (x) to be valid, it must produce the same reduced equation when we begin

with the mKdV equation iterated up once in m. That is

ˆx = xˆx − r ˆx

ˆx − r ˆx(4.8)

80



must lead to the same reduction, with possible conditions placed on r. We note that

x = f (x) so ˆx = f (x) = f (f (x)) and so (4.8) is equivalent to

f (f (x)) = f (x)

f (x)

−rf (f (x))

f (f (x)) − rf (x)

4.3.1 f (x) = xξ

The first reduction we consider is f (x) = xξ, ξ constant, in (4.7) so that

¯xξ = xx1−ξ − r

1 − rx1−ξ (4.9)

and we use the same f (x) in (4.8) to get

¯xξ = x

xξ(1−ξ) − r

1 − rxξ(1−ξ)

1/ξ(4.10)

The two expressions for ¯xξ, (4.9) and (4.10), agree if ξ = 1 but this leads to a linear

equation. Another solution is ξ = −1 and r = r. The latter condition on r dictates

through (4.5) that r = k1kl2, where ki are constant, so that the final form of the

reduced equation is

¯xx = k1kl2x2 − 1k1kl2 − x2

(4.11)

which is a special case of a q-discrete Painleve III equation (qPIII) found by [33].

This equation was already obtained in [86] as a reduction of the lattice sine-

Gordon equation (LSG). The advantage of the reduction presented above is that

it comes with a Lax Pair (see §4.4.2). As a point of interest we mention that the

LMKdV can be transformed to the LSG by allowing x → 1/x.

4.3.2 f (x) = ¯xξ

Now consider f (x) = ¯xξ the same analysis as above shows that, again, ξ = 1 or ξ =

−1 will lead to valid reductions. When ξ = 1, we must set log r = k1l + k2 + k3(−1)l

81



and, after introducing y = ¯x/x, we are left with

yy =1 − ry

y(y − r). (4.12)

The same equation as (4.12) was found in [86] where the equation was identified as

either qP II or qP II I , depending on whether k3 = 0.

Now take the case when ξ = −1, this time (4.7) becomes

¯xx =¯x¯xxx

¯xx=

r ¯xx − 1

r − ¯xx

whereupon setting y = ¯xx we find

yy = y ry − 1r − y

. (4.13)

To find the required form of the parameter functions we must compare this to (4.8)

with the same y substituted

¯yy = ¯y1 − r ¯y¯y − r

Clearly, the equivalence between these two mappings is satisfied by taking r as for

the case when ξ = +1. Equation (4.13) is actually equivalent to a special case of

(4.12) and was also derived in [86] but from the lattice Sine-Gordon equation rather

than the lmKdV.

4.3.3 f (x) = ¯x

We let f = ¯x and, on substituting w = ¯x/x, we have

ww = 1 − rww − r

. (4.14)

Here logr = k1l + k2 + k3 jl + k4 j

2l, ki constants and j3 = 1. This equation was

shown to be a qP II when k3 = k4 = 0 [87] or a qP V in the general case [88].

82



4.3.4 f (x) = 1/ ¯x

Specifying f = 1/ ¯x leads to what appears to be an irreducible, fourth order, inte-

grable difference equation, namely

¯xx =xx− r

1 − rxx(4.15)

where again logr = k1l + k2 + k3 jl + k4 j

2l and j3 = 1.

4.3.5 f (x) = xl+4

The reductions of orders higher than third all lead to equations that are not reducible

to a second order form. The next reduction to consider is f = xl+4 which becomes

¯yyyyy =1 − ryyy

yyy − r(4.16)

with y = ¯x/¯x, and log r = k1l + k2 + k3(−1)l + k4 cos( lπ2

) + k5 sin( lπ2

), ki= constants.

4.3.6 f (x) = 1/xl+4

Using f = 1/xl+4 allows the reduction of the lmKdV to

¯yyy = yyryy − y

ry − yy(4.17)

where r is the same as in the previous example but y = ¯x¯x.

Arbitrarily high order equations can be generated in this manner.

4.4 Lax Pairs for the Reduced Equations

In this section, we deduce Lax pairs for the q-discrete Painleve equations derived in

the previous section by applying the observations obtained from the multi-dimensional

83



and on the other hand,

¯θ

= ¯t

1

¯x

=

¯t

t

v

But θ

= ( N

)−1M θ and ¯θ

= (N

)−1LLθ. Thus

¯t

N

= LLM −1t

N

(4.20)

We now use (5.2b) as a guide and try a general N that has the same form as LLM −1,

where by the same form we mean that it contains the same powers of ν .

LLM −1 =1

ν 2

− µ2

ν 2 + λλx/¯x − µλ¯x/x− µλx/x ν (µ/x− λ¯x/xx− λ/x) + 1ν

µλλ/¯x

ν (µx− λx− λxx/¯x) + 1ν µλλ¯x ν 2 + λλ¯x/x− µλx/x− µλx/¯x Since the prefactor cancels in the compatibility condition (4.22), we take N to be

N =

ν 2a2 + a0 νb1 + b0/ν

νc1 + c0/ν ν 2d2 + d0

(4.21)

where the coefficients ai, bi, ci and di are functions of l only. Before we continue,

we will assume ν (n) = qn and keep ν as the spectral variable in L and N . As we

are seeking a Lax Pair with coefficient matrices N , L that depend on a spectral

parameter ν , x should be independent of ν . Hence we take x = x in the following.

Now the coefficients of the various powers of ν in N are determined by the

compatibility condition LN = NL (4.22)

which is the compatibility condition of the system (4.18). Going through the calcu-

lations in detail would be somewhat tedious so only a guide will be given here. The

compatibility condition gives a total of ten equations, three in each of the diagonal

entries and two in the off-diagonal entries. The equations in the diagonal entries at

85



order ν 2 and ν −2 are solved in a straightforward manner, yielding

a2 = constant

d2

= constant

b0 =k1xσ

c0 =k1x

σ

where k1 is a constant and σ = ql. The remaining six equations read as follows:

a0 − a0 = λ(xb1 − c1/qx) (4.24a)

d0 − d0 = λ(c1/x− xb1/q) (4.24b)

b1x − b1x = λ(a2 − d2/q) (4.24c)

c1/x− c1/x = λ(d2 − a2/q) (4.24d)

a0 − d0/q =k1λσ

(x

qx− x

x) (4.24e)

d0 − a0/q =k1λσ

(x

qx− x

x) (4.24f)

To solve these, use (4.24c) to replace b1x in (4.24a), and the resulting expression to

replace a0 in (4.24e). Now do the same with (4.24d) and (4.24b) in (4.24f), then

solve these two equations for a0 and d0 to find

a0 = − k1x

λσx− λb1x− λ2a2 (4.25a)

d0 = − k1x

λσx− λc1

x− λ2d2(4.25b)

One can now use (4.25a) and (4.25b) in equations (4.24a) and (4.24b), replace b1

and c1 via (4.24c) and (4.24d), then solve the remainder for b1 and c1. All this

reduces to

a0 =a2λλx

¯x− k1x

σ(

1

λ¯x+

1

λx) (4.26a)

d0 =d2λλ¯x

x− k1

σx(

¯x

λ+

x

λ) (4.26b)

b1 =k1

λλσ ¯x− a2

x(λ + λ

x¯x

) (4.26c)

c1 =k1 ¯x

λλσ− d2x(λ + λ

¯x

x) (4.26d)

86



Finally, these calculated values should be substituted back into equations (4.24a)–

(4.24f). On doing this and making the substitution y = ¯x/x, we find that one of the

two following forms of qPII arises in each case

yy =1

y

k1λy − a2λλ¯λ2qσ

qk1¯λ − d2λ2λ¯λσy

(4.27)

yy =1

y

k1¯λqy − a2λ2λ¯λσ

k1λ − d2λλ¯λ2qσy(4.28)

Equations (4.27) and (4.28) can be reconciled by setting log λ = k2+k3(−1)l+ l2

log q,

with constant k2 and k3, which gives the same form of qPII that was given earlier

(and found in [86]) by a reduction from the LMKdV equation. Hence we have

calculated a Lax Pair for qPII, which explicitly takes the form

θ(l + 1, n) = L

l, ν (n)

θ(l, n) (4.29a)

θ(l, n + 1) = N

l, ν (n)

θ(l, n) (4.29b)

where

L =

xx

− λνx

−λxν

1

, (4.30)

N = a2ν 2 + a2λλx¯x − k1xσ ( 1λ¯x + 1λx) ν [ k1λλσ ¯x − a2x (λ + λ xx)] + k1νσx

ν [ k1 ¯xλλσ

− d2x(λ + λ ¯xx

)] + k1xνσ

d2ν 2 + d2λλ¯xx

− k1σx

( ¯xλ

+ xλ

)

(4.31)

4.4.2 Lax Pair for qPIII

The next reduction to be considered is that taking LMKdV to qPIII , i.e., x = 1/x.

The reciprocal in the latter reduction introduces a difference in the method used to

find the corresponding Lax Pair. We now have

θ

= t

1

x

= t

1

1/x

= t

/x

x

1

and

θ

= t

1

x

=xt

t

0 1

1 0

v

87



So this time the suggested form of N is the same as

0 1

1 0

LM −1 and, as

such, we choose

N = a/ν b0 + b2/ν 2

c0 + c2/ν 2 d/ν

(4.32)

In this case the compatibility condition contains eight equations that are solved in

a similar way to before, and these lead to

a = −λk1xx− k2x

λσx(4.33a)

b0 = k1x (4.33b)

b2 =k2σx (4.33c)

c0 =k3x

(4.33d)

c2 =k2x

σ(4.33e)

d = −λk3xx

− k2x

λσx(4.33f)

where k2, k1 and k3 are all constants, λ = δq−l where δ = constant and σ = ql.

The final form of the qPIII equation obtained from the compatibility conditions

is

x¯x =µ1qlx2 + µ2

µ3ql + x2(4.34)

which is a non-autonomous equation with three free parameters µi. The Lax Pair

is of the form

θ(l + 1, n) = L

l, ν (n)

θ(l, n) (4.35a)

θ(l, n + 1) = N l, ν (n)θ(l, n) (4.35b)

where L as before is given by

L =

xx

− λνx

−λxν

1

, (4.36)

88



and

N =

− 1

ν (λk1xx + k2x

λσx) k1x + k2

ν 2σx

k3x

+ k2xν 2σ

− 1ν

(λk3xx

+ k2xλσx

)

(4.37)

4.4.3 Lax Pair for qPV

Lastly, a Lax Pair for qPV (see equation (4.14)) is presented. Following the previous

analysis, we begin with N of the same form as LLLM −1 or

N =

a1ν 2 + a0 + a2/ν 2 b1ν + b0/ν

c1ν + c0/ν d1ν 2 + d0 + d2/ν 2

(4.38)

After similar processes to those used earlier, we arrive at the following, where

log(T 2) = K 1 + K 2(−1)l, K i constant, is a function of period two, and σ = ql

as before.

a0 =T 2x

σ ¯x(

¯x¯λλx

+¯x ¯x

λλxx+

1¯λλ

) + a1(λλx¯x

+ λ¯λxx

¯x ¯x+ λ¯λ

x¯x

)

a1 = constant

a2 =T 2

σd0 =

T 2 ¯x

σx(

x

λ¯λ¯x+

xx

λλ¯x ¯x+

1

λ¯λ) + d1(λλ

¯x

x+ λ¯λ

¯x ¯x

xx+ λ¯λ

¯x

x)

d1 = constant

d2 =T 2σ

b0 = −T 2 ¯x

σx(

1¯λ ¯x

+1

λx+

x

λx¯x) − a1

λλ¯λ¯x

b1 = − T 2

λλ¯λσ ¯x− a1

¯x(λ

¯x

x+ λ

x ¯x

x¯x+ ¯λ

x¯x

)

c0 = − T 2x

σ ¯x(

¯x¯λ

+x

λ+

x¯x

λx) − d1λλ¯λ ¯x

c1 = − T 2 ¯x

λλ¯λσ− d1 ¯x(λ

x¯x

+ λx¯x

x ¯x+ ¯λ

¯x

x)

We also find that log λ = k1 + k2 jl3 + k3 j

l3 − ql/3, j33 = 1 and the form of the

89



resulting evolution equation is

yy =qT 2

¯λy + a1λ2λ¯λ

¯λν

T 2λ + qd1λλ¯λ¯λνy

(4.39)

where we have made the substitution y = ¯x/x.

In this case, the Lax pair takes the form

θ(l + 1, n) = L

l, ν (n)

θ(l, n) (4.40a)

θ(l, n + 1) = N

l, ν (n)

θ(l, n) (4.40b)

where L as before is given by

L = x

x −λ

νx−λxν

1 , (4.41)

and we write N as

N = N 2 ν 2 + N 1 ν + N 0 +N −1

ν +

N −2ν 2

(4.42)

where

N 2 =

a1 0

0 d1

N 1 =

0 − T 2λλ¯λσ ¯x

− a1¯x

(λ¯xx

+ λx ¯xx¯x

+ ¯λx¯x

)

− T 2 ¯x

λλ¯λσ− d1 ¯x(λ x

¯x+ λ x¯x

x¯x+ ¯λ ¯x

x) 0

N 0 =

T 2xσ ¯x

( ¯x¯λλx

+ ¯x ¯xλλxx

+ 1¯λλ

) + a1(λλx¯x

+ λ¯λxx¯x ¯x

+ λ¯λ x¯x

)

0

0¯T 2 ¯xσx

( x

λ¯λ

+ xx

λ¯λx

¯x

+ 1¯λ¯λ

) + d1(λλ ¯xx

+ λ¯λ ¯x ¯xxx

+ λ¯λ¯xx

)

N −1 =

0 −T 2 ¯xσx

( 1¯λ¯x

+ 1λx

+ xλx¯x

) − a1λλ¯λ¯x

− T 2xσ ¯x

(¯x¯λ

+ xλ

+ x¯xλx

) − d1λλ¯λ ¯x 0

and

N −2 =

T 2σ

0

0 T 2σ

.

90



4.5 Discussion

In this chapter we have presented a new Lax Pair for a lattice, non-autonomous,

modified Korteweg de Vries equation and shown that it forms a consistent multi-

dimensional system when considered together with its Lax pair. We also gave reduc-

tions of this non-autonomous LMKdV to q-difference Painleve equations and found

the Lax Pairs corresponding to those reduced equations. A notable feature of these

results is that they provide 2 × 2 Lax pairs for the first time for these versions of

qPII, qPIII , and qPV.

It is worth noting that only one simple form of reduction was investigated here.

It remains to be seen whether other types of reductions, that is other forms of f (x)

in (4.7), can be used with the LMKdV or other lattice equations.

We note that, so far, there appears not to be a direct method for reducing the

lattice Lax pair L, M to the ordinary difference equation’s Lax pair L, N . There is

a jump in our process of finding N after reduction. The main obstacle is that it is

not known whether equations of the form (5.2b) can be solved to find N directly.

Instead, we have chosen to use the form of the equation to motivate the dependence

of N on the spectral parameter ν and then used the compatibility conditions to

deduce the entries of N .

A feature of the Lax pairs we deduce is that they share the same L. We note

here that the calculation of a series of Lax Pairs is also possible for other reductions,

including the higher-order difference equations found in §2. These would also share

the same L matrix. This is analogous to the case of integrable differential-equation

hierarchies and suggests the existence of a hierarchy for each of the reductions we

have studied here. An open problem is to find reductions of lattice equations to

infinite hierarchies of q-difference equations along with their Lax pairs. It would

be eventually interesting to find reductions from lattice equations to the q-Garnier

hierarchy constructed by Sakai in [35].

91



It is striking to note also that very little information is known about the generic

solutions of q-Painleve equations. To our knowledge, Birkhoff ’s theory of linear q-

difference equations has not been applied to deduce information about the solutions

of q-Painleve equations. The question of the global properties of solutions remains

completely open.

92



Chapter 5

Hierarchies of equations

5.1 Introduction

We have seen in chapter 4 that reductions constitute a natural connection between

lattice equations and ordinary difference equations. The LMKdV equation is de-

autonomized by allowing p and r to depend on l and m and there are known reduc-

tions from the non-autonomous LMKdV equation to q-discrete forms of the second,

third and fifth Painleve equations, denoted qPII, qPIII , and qPV respectively [86, 2].

Different types of these reductions are possible, one of the simplest of which is to

set xl,m+1 = xl+d,m for some positive integer d. In fact the reductions that take

the LMKdV equations to qPII and qPV are of this type with d = 2 and d = 3 re-

spectively. There apparently exist an infinite series of such reductions that result in

equations of arbitrary order, which suggests the existence of a hierarchy of equations.

In a recent paper [2] we established a connection between these non-autonomous re-

ductions and reductions of a Lax pair for the LMKdV equation itself. In this way,

Lax pairs for non-autonomous versions of qPII, qPIII , and qPV with multiple free

parameters were discovered.

Here we find Lax pairs for higher order equations and we thereby lay the ground-

93



work for a hierarchy of equations. Two hierarchies are shown to exist, at the base

of one lies qPII and qPV, while qPIII lies at the base of the other.

While there is a relatively large body of literature focussing on continuous

Painleve hierarchies [89, 114, 115] and some results concerning hierarchies of d-

discrete nonlinear equations [99, 89], the problem of q-discrete hierarchies has been

more elusive. Although a hierarchy of integrable nonlinear q-difference equations was

found in [98], we believe that the results presented in this work are first example of

such a hierarchy found by expansions of Lax pairs.

The chapter is organized as follows. In section 5.2, we derive the formulas used

to calculate quantities that exactly describe an equation in one of the hierarchies.

These are given in terms of the same quantities describing the equation at a lower

order and so we thereby obtain a recursive method of finding the hierarchies. We

go further to purport a general formula that yields those quantities for any member

of either hierarchy, and thus any equation in the hierarchies with its Lax pair. In

section 5.3, we clarify the application of the formulas found in section 5.2 and use

them to confirm a known result. In section 5.4, we derive new equations and Lax

pairs. We conclude the chapter with a discussion where we also point out some open

problems.

5.2 How to Construct the hierarchy

In this section we will establish the existence of two hierarchies of nonlinear, inte-

grable, ordinary q-difference equations that are each obtainable from the LMKdVequation via a reduction. In section 5.2.1, the procedure for constructing the hierar-

chies will be derived in relation to the first hierarchy, which corresponds to reductions

of the type xl,m+1 = xl+d,m for some integer d. Subsequently, in section 5.2.2 we

shall outline an analogous process that leads to the second hierarchy corresponding

to reductions of the type xl,m+1 = 1/xl+d,m.

94



We establish the existence of the hierarchies by developing formulas that con-

struct a member of the hierarchy from the preceding, lower order member. However,

rather than iterating the equation or terms in the Lax pair directly, as has been the

procedure used for some other systems [99], we will derive formulas for iterating a

set of coefficients, introduced in equation (5.14), that describe the Lax pairs for the

equations in the hierarchy.


xl+d,m

Begin with the linear problem

θ(l + 1, n) = L(l, n)θ(l, n),

θ(l, n + 1) = N (l, n)θ(l, n).(5.1)

whose compatibility condition is L(l, n + 1) N (l, n) = N (l + 1, n) L(l, n). Hereafter

we adopt the notation θ = θ(l + 1, n) and θ = θ(l, n + 1). Now set

L = x/x −ν/(λx)

−ν x/λ 1

, (5.2a)

N =

a0 + a2ν 2 + ... + a2ρν 2ρ b1ν + b3ν 3 + ... + b2ρ±1ν 2ρ±1

c1ν + c3ν 3 + ... + c2ρ±1ν 2ρ±1 d0 + d2ν 2 + ... + d2ρν 2ρ

(5.2b)

where k is associated with the spectral variable n such that ν = ν 0qn, and x, λ and

all of ai, bi, ci, di are functions of l alone. The diagonal entries of the N matrix in

the Lax pair contain only even powers of ν , including a term constant in ν , up to

ν 2ρ where ρ is a positive integer. The off-diagonals of N contain only odd powers of

ν up to ν 2ρ±1 depending on which part of the hierarchy we are considering.

Compatibility occurs when LN = N L. It is not difficult to show that the

95



compatibility condition can be written as follows

ai = ai +1

λ(xbi−1 − q

ci−1x

), i even (5.3a)

xbi = xbi +1

λ(ai−1 − qdi−1), i odd (5.3b)

cix

=cix

+1

λ(di−1 − qai−1), i odd (5.3c)

di = di +1

λ(

ci−1x

− qxbi−1), i even (5.3d)

corresponding to entries (1, 1), (1, 2), (2, 1) and (2, 2) respectively. Some equiva-

lences may be found between equations (5.3) if, at this point, we introduce the

quantities

Ai =

ai, i even

xbi, i odd

(5.4)

Di =

di, i even

ci/x, i odd

(5.5)

so that equations (5.3) become

x¯x

Ai−1 = λ(Ai − Ai) + qDi−1, i even (5.6a)

Ai−1 = λ(x¯x

Ai − Ai) + qDi−1, i odd (5.6b)

Di−1 = λ(¯x

xDi −Di) + qAi−1, i odd (5.6c)

¯x

xDi−1 = λ(Di − Di) + qAi−1, i even (5.6d)

In equations (5.6) we may substitute

X i =

¯x

x

1−(−1)i

2

=

¯x/x, i odd

1, i even

so that either equation (5.6a) or (5.6b), with i even or odd respectively, will become

Ai−1X i−1

= qDi−1 + λ(Ai

X i−Ai), ∀i (5.7)

96



and similarly equations (5.6c) and (5.6d), with i odd or even respectively, become

Di−1X i−1 = qAi−1 + λ(DiX i − Di), ∀i (5.8)

By repeated use of equations (5.7) and (5.8) respectively, we arrive at the following

Ai = X i[qDi −m

j=i+1

λ j−i(A j − qD j)] (5.9a)

Di =1

X i[qAi −

m j=i+1

λ j−i(D j − qA j)] (5.9b)

Where m is equal to the greatest degree of the polynomials in k located in the entries

of the N matrix (5.2b), i.e. m is either 2ρ or 2ρ + 1. Now add q/X i × (5.9a) to X i× (5.9b) so that

q

X iAi + X iDi = q2Di + qAi +

m j=i+1

(q2 − 1)λ j−iD j (5.10)

However, at i = 0, (5.6) shows that both A0 and D0 are constant, meaning that

(5.10) at i = 0 becomes

−D0 =

m

j=1 λ jD j

which we can rearrange so that

−D1 = D0/λ +m

j=2

λ j−1D j (5.11)

Similarly q(5.9a) + (5.9b) gives us an expression for A1

−A1 = A0/λ +m

j=2λ j−1A j (5.12)

These expressions for A1 and D1, (5.11) and (5.12), can be substituted back into

(5.7) and (5.8) to find expressions for A2 and D2 in terms of Ai, Di with i > 2. We

can continue this process to successively calculate all the terms in the Lax pair, Ai

and Di, thus resolving the Lax pair for a particular value of m. However, in the

interest of establishing the existence of a hierarchy of equations, we will proceed to

97



derive a formula for calculating successive iterates from previous ones. To do this

we first rewrite (5.9a) as

−¯

Ai + qX iDi + X i

m

j=i+1 λ

j

−i

(qD j −A j) = 0 (5.13)

We aim to calculate each of the quantities in the Lax pair, Ai, Di, in terms of the

remaining quantities, A j, D j ∀ j > i and A0,D0. In general any Ai of interest might

be found in terms of all A j and D j, ∀ j > i. However, when the calculations are

performed, it is observed that terms Ai only depend terms A j (not D j) so it is

conjectured that we can write any Ai or Di as

−Ai = αi0A0 +

m j=i+1

αi jA j (5.14a)

−Di = δi0D0 +m

j=i+1

δi jD j (5.14b)

Where we have introduced a series of coefficients αi j and δi j that need to be found.

Substitute the expansion (5.14a) into (5.9a), noting that A0 = constant, to get

αi0A0 +

m

j=i+1αi jA j + qX iDi + X i

m

j=i+1λ j−i(qD j − A j) = 0

which, upon exploiting (5.9a) again to replace A j, becomes

αi0A0 − qX iδ

i0D0 +

m j=i+1

αi jX j

mk= j+1

λk− j(qDk − Ak)

+m

j=i+1

{D jq[X i(λ j−i − δi j) + αi jX j] − λ j−iA jX i} = 0

Now we may rearrange the double sum to arrive at the following

αi0A0 − qX iδ

i0D0 + Di+1q[X i(λ − δii+1) + αi

i+1X i+1] − λAi+1X i

+m

j=i+2

{D jq[X i(λ j−i − δi j) + αi jX j +

j−1k=i+1

λ j−kX kαik]

− A j(λ j−iX i +

j−1k=i+1

λ j−kX kαik)} = 0 (5.15)

98



A repeat of this process beginning with (5.9b) brings us to

δi0D0 − qαi0

X iA0 + Ai+1q[

1

X i(λ − αi

i+1) +δii+1X i+1

] − λDi+1

X i

+m

j=i+2

{A jq[ 1X i

(λ j−i − αi j) + δi j

X j+

j−1k=i+1

λ j−kδikX k

]

− D j(λ j−i

X i+

j−1k=i+1

λ j−kδikX k

)} = 0 (5.16)

To calculate the next sets of coefficients αi+1 j and δi+1 j , j = i + 2,...,m, we must

combine equations (5.15) and (5.16) in the correct way. We claim that we can

eliminate the quantities Di by adding q(5.16)+δi0/(δi0X i) (5.15), which tallies to

A0(αi0δi0

X iδi0− q2

αi0

X i) + Ai+1{q2[

1

X i(λ − αi

i+1) +δii+1X i

] − λδii+1δi0

}

+m

j=i+2

A j{q2[1

X i(λ j−i−αi

j)+δi jX j

+

j−1k=i+1

λ j−kδ jkX k

]−(λ j−iδi0

δi0+

j−1k=i+1

λ j−kX kδi0α jkX iδi0

)} = 0

(5.17)

It is not obvious that every Di should be canceled out in the above sum but this

occurs in every calculation performed to date and we conjecture that it is always

the case. From here we can make −Ai+1 the subject and so find the sought after

coefficients

− Ai+1 = A0αi0δi0 − q2αi

0δi0q2[δi0(λ− αi

0) + Xi

Xi+1δi0δii+1] − λX iδi0

+m

j=i+2

A j{q2[δi0(λ j−i − αi

j) +δi0δ

ij

Xj+

j−1k=i+1

λj−kδi0δjkXi

Xk]− (λ j−iδi0X i +

j−1k=i+1 λ j−kδi0α jk)

q2[δi0(λ − αi0) + Xi

Xi+1δi0δii+1]− λX iδi0

}

(5.18)

Comparing (5.18) with (5.14a) shows that

αi+10 =

αi0δi0 − q2αi

0δi0q2δi0[λ − αi

i+1 + Xi

Xi+1δii+1] − λX iδi0

(5.19a)

αi+1 j =

1Gi(α,δ,X)

{q2δi0[λ j−i − αi j +

δijXi

Xj+

j−1k=i+1

λj−k δikXi

Xk]

−δi0(λ j−iX i + j−1

k=i+1 λ j−kαikX k)}

(5.19b)

99



Where Gi in (5.19b) is the same as the denominator in (5.19a). In fact we shall say

αi+10 =

H i0(α, δ, X)

Gi(α, δ, X)(5.20a)

αi+1 j = H

i j(α, δ, X)

Gi(α,δ, X)(5.20b)

The H and G quantities in (5.20) are defined by comparison with (5.19) where we

have introduced the bold face notation α to signify all α, α, etc, with any superscripts

and subscripts.

Naturally, we must also repeat the operations to find an expression for the other

coefficients δi j, this begins with adding q(5.15)+αi0X i/αi

0 (5.16), and leads to

δi+10 =δi0αi

0 − q2δi0αi0

q2αi0[λ − δii+1 + Xi+1

Xiαii+1]− λαi0

Xi

(5.21a)

δi+1 j =1

Gi(δ,α,1/X){q2αi

0[λ j−i − δi j +αijXj

Xi+

j−1k=i+1

λj−kαikXk

Xi]

−αi0(λ j−i/X i +

j−1k=i+1

λj−kδik

Xk)}

(5.21b)

Notice that the quantities H and G from (5.20) arise again in equations (5.21) but

this time as

δi+10 = H i0(δ,α,

1X)

Gi(δ,α, 1X

)(5.22a)

δi+1 j =H i j(δ,α, 1

X)

Gi(δ,α, 1X

)(5.22b)

Importantly, since α1 j = δ1 j , (5.20) and (5.22) indicate that αi

j(λ, x) = δi j(λ, 1x

) for

i > 1. Hence, we only need to calculate the coefficients αi j in practice as δi j follow

from these results.

Because these coefficients exactly describe a Lax pair and an associated nonlinearequation, we have shown that this system does indeed constitute a hierarchy by

constructing a general operation that takes the members at any level of the hierarchy

to the next level. To find a particular Lax pair in the hierarchy, we truncate the

series at some point Am, Dm say, and use the coefficients to calculate each of the

terms Ai, Di that appear in the N matrix of the Lax pair, the L matrix is always the

100



same. The equation associated with any Lax pair can be found via the compatibility

condition. We may also continue to find higher order members of the hierarchy by

subsequently reinstating some of the terms Aι, Dι with ι > m and calculating the

coefficients needed to describe those terms, αi j and δi j, through equations (5.19) and

(5.21).


1/xl+d,m

The formulas for the coefficients that were found in the preceding section corre-

sponded to equations that can be obtained from the LMKdV equation via the re-

duction x = xl+d. However, in [2] it was shown that reductions of the type x = 1/xl+d

can also be used and that these reductions lead to q-discrete Painleve equations as

well. The hierarchy of equations that springs from this type of reduction has Lax

pairs that are very similar to the ones used in section 5.2.1 and fit easily into the

present framework. The main difference between the two sets of Lax pairs can be

described in terms of the spectral parameter k. The Lax pairs in section 5.2.1 allhave the following form:

N =

a0 + a2k2 + ... + a2ρk2ρ b1k + b3k3 + ... + b2ρ±1k2ρ±1

c1k + c3k3 + ... + c2ρ±1k2ρ±1 d0 + d2k2 + ... + d2ρk2ρ

Observe that the terms that contain the lowest power of k, that is terms that are

constant in k, appear in the diagonal entries of N . Also note that the other half of

the Lax pair, the L matrix from equation (5.2a) remains the same for both types of

reduction.

Moving now to the hierarchy associated with reductions of the type x = 1/xl+d,

we find that the associated Lax pairs have a form similar to the former case, except

here the lowest powers of k appear in the off diagonal entries. This can be achieved

101



simply by removing the constant terms from the diagonal entries.

N =

a2k2 + ... + a2ρk2ρ b1k + b3k3 + ... + b2ρ±1k2ρ±1

c1k + c3k3 + ... + c2ρ±1k2ρ±1 d2k2 + ... + d2ρk2ρ

We can find the hierarchy that arises from this case in a congruent manner to

the last with only minor alterations. The differences here arise because A1 is now

the first term in the series but it is not a constant, as was A0 in the previous case.

Instead A1 = β 1xx, β 1 = constant, which introduces additional factors of X 1 = ¯x/x

into the equations after (5.14a). Following the same procedure as in section 5.2.1,

it is not difficult to show that in this case the formulas analogous to (5.19) are

αi+11 = α

i1δ

i1X

2i − q

2

αi1δ

i1

q2δi1[λ− αii+1 + Xi

Xi+1δii+1] − λX 1X iδi1

(5.23a)

αi+1 j =

1Gi{q2δi1[λ j−i − αi

j +δijXi

Xj+

j−1k=i+1

λj−k δikXi

Xk]

−δi1(λ j−iX 1X i + j−1

k=i+1 λ j−kαikX 1X k)}

(5.23b)

Where Gi in (5.23b) is the same as the denominator in (5.23a). We will present

the first few equations in this hierarchy in section 5.4.2

5.2.3 General Coefficients

It is conjectured that all the coefficients for the hierarchy of equations that arise

from reductions of the type xl,m+1 = xl+d,m are given by the following equations

αk j =

j−ki1=0

j−k−I 1i2=0

...

j−k−I k−2ik−1=0

k−2h=1,h odd

h

X I h−I k−1

k−2g=0,g even

g

X I g−I k−1

(−1)kk−1f =0

f

λif (5.24a)

αk0 =

k−1h=0

hλ−

1 k−2g=0

gX k+g+1

−1

(5.24b)

Where I k =k

ς =1 iς , i0 = j − k − I k−1,

X i =

¯x

x

1−(−1)i

2

=

¯x/x, i odd

1, i even

102



and we use the notationf

λ = λl+f .

These formulas can be used to find any coefficient of interest, which vastly de-

creases the number of calculations required to find an equation of any order in thehierarchy.

We obtain similar results for the coefficients for the hierarchy corresponding to

reductions of the type xl,m+1 = 1/xl+d,m.

αk+1 j+1 =

j−ki1=0

j−k−I 1i2=0

...

j−k−I k−2ik−1=0

k−2g=0,g even

g

X I g−I k−1

k−2h=1,h odd

h

X I h−I k−1

(−1)kk−1f =0

f

λif (5.25a)

αk+11 =

k−1h=0

h

λ−1 k−2

g=0

g

X k+g+1 (5.25b)

These coefficients αk+1 j+1 are equal to δk j from the first hierarchy.

5.3 A Known Example

In this section we will implement the formulas (5.24) to explicitly find a known

example. The procedure runs as follows:

- First, we decide how many terms we will keep in the Lax pair, i.e. we decide

which Ai, and Di will be nonzero, up to i = m say.

- Second, use equation (5.24) to calculate all the coefficients αi j up to αm−1

m . It

is necessary to calculate every αi j with j

≤m and i

≤m

−1 in order to specify

the Lax pair.

- Third, calculate the terms Ai and Di from equations (5.14a) and (5.14b),

noting that any δi j is equal to αi j with X i, X i, X i, . . . replaced with 1/X i,

1/X i, 1/X i, . . . . We may then find the corresponding nonlinear equation using

the compatibility conditions (5.7) and (5.8).

103



For our example we shall retain only those terms Ai, Di with 0 ≤ i ≤ 3, which

causes there to be two terms in each entry of the N matrix of the Lax pair (see

(5.2b)). A Lax pair of this form was already presented in [2] where it was shown to

correspond to qP II , we expect the same to occur here.

The next step is to calculate the coefficients αi j , δi j up to i = 2 and j = 3. We

begin with the coefficients α1 j and δ1 j , for which inspection of equations (5.11) and

(5.12) indicates

α1 j = δ1 j = λ j−1 (5.26)

Directly from (5.24) we find

α20 = −1/(λλX 1) (5.27a)

α23 = λ + λ/X 1 (5.27b)

We have now calculated all the coefficients needed for the present example but we

will list the next four as well, for future reference.

α24 = λ2 + λλ/X 1 + λ2 (5.28a)

α25 = λ3 + λ2λ/X 1 + λλ2 + λ3/X 1 (5.28b)

α26 = λ4 + λ3λ/X 1 + λ2λ2 + λλ3/X 1 + λ4 (5.28c)

α27 = λ5 + λ4λ/X 1 + λ3λ2 + λ2λ3/X 1 + λ4λ + λ5/X 1 (5.28d)

At this point we use the coefficients to calculate the values of the nonzero terms

in the N matrix. Since A0 and A3 are at the ends of the sequence, we can calculate

their values directly from (5.7), using the appropriate values of i in that equation.

Trivially, these are found to be A0 = a0 = constant and A3 = xb3 = T 2σx/x where

T 2 is an arbitrary period two function of l and σ = ql. The lower case a0 and b3 are

the original variables in the N matrix, see (5.2b). Using (5.14a)

−A2 = α20A0 + α2

3A3

= − a0x

λλ¯x+ (λ +

λx¯x

)T 2σx

x(5.29)

104



We then use the coefficients from the previous step to calculate A1

−A1 = α10A0 + α1

2A2 + α13A3

= a0(1

λ +x

λ¯x) −λλT 2σx

¯x (5.30)

Finally, we can obtain the related equation by substituting these values into (5.7)

at i = 3 whence we recover qPII as expected. The form of the equation is

yy =1 − T 2ry

y(γy − T 2r)(5.31)

where log r = γ 0 + γ 1(−1)l − ql/2, γ, γ i = constant, and y = ¯x/x. Actually this

version of qPII contains more parameters than those found in [86, 2] as γ and T 2,which are described after the Lax pair below, were not present in those papers. The

corresponding Lax pair is

L =

x/x −ν/(λx)

−ν x/λ 1

,

N =

a0 + ν 2 a0xλλ¯x

− (λ + λx¯x

)T 2σxx

ν 2 ν (−a0x

( 1λ

+ xλ¯x

) + λλT 2σ¯x

) + ν 3 T 2σx

ν (

−d0x( 1

λ+ ¯x

¯λx

) + λλT 2σ ¯x) + T 2σxν 3 d0 + ν 2 d0 ¯x

λ¯λx −

ν 2(λ + λ¯xx

)T 2σxx

The terms in the N matrix are related to those in (5.31) by γ = d0/a0, T 2 is an

arbitrary, period-two function of l and r = λλ¯λσ/a0 where σ = ql. The spectral

parameter is n and it enters the Lax pair via ν = ν 0qn.

5.4 Higher Order Equations

Now that the formulas for finding all the equations in the hierarchy have been

derived and their use explained, we will write down some higher order equations

and their associated Lax pairs. Section 5.4.1 will deal with equations obtained from

the LMKdV equations via reductions of the type xl,m+1 = xl+d,m and 5.4.2 will deal

with the type xl,m+1 = 1/xl+d,m.

105



5.4.1 Equations Corresponding to Reductions of the Type

xl,m+1 = xl+d,m

This subsection pertains to higher order equations that can be obtained from the

LMKDV equation by using a reduction of the type x = xl+d where d is some positive

integer. The coefficients, α3 j , that will be required for all of the equations presented

in this section are calculated using (5.24) and are listed below.

α30 = 1/(λλ¯λX 1)

α34 = λ + λX 1 +

¯λX 1

X 1

α35 = λ2 + λλX 1 +

λ¯λ

X 1+

λ¯λX 1

X 1+ λ2 + ¯λ2

α36 = λ3 + λ2λX 1 +

λ2 ¯λX 1

X 1+ λλ2 +

λλ¯λ

X 1

λ¯λ2 + λ3X 1 +λ2¯λX 1

X 1+ λ¯λ2X 1 +

¯λ3X 1

X 1

The Lax pair and associated equation that is achieved by truncating the series Ai

at A4 was derived in [2]. The equation is qPV, which can be obtained from the

LMKdV equation via the reduction x = ¯x [86, 2].

qPV : ww =1 + T 2rw

T 2r + γw(5.34)

Where w = ¯x/x, γ = constant, T 2 is an arbitrary period-two function and log r =

γ 0 + γ 1 jl3 + γ 2 j

2l3 − ql/3, with γ ι = constant. Noting that X 1 = ¯x/x, we can use the

coefficients αi j to find the Lax pair for this qPV equation through (5.14a). The Lax

pair lies below and the relationships between the terms in the Lax pair and those

106



in the equation (5.34) follow.

L =

x/x −ν/(λx)

−ν x/λ 1

(5.35)

N 11 = a0 + ν 2a0(x

λλ¯x+

x

λ¯λ ¯x+

xx

λ¯λ¯x ¯x)

+T 2σ(λλ¯x

x+

λ¯λx¯x

x ¯x+

λ¯λx¯x

) + ν 4T 2σ

N 12 = −νa0(1

λx+

x

λx¯x+

x¯λ¯x ¯x

) − νT 2σλλ¯λ

¯x− ν 3a0

λλ¯λ ¯x+ T 2σ(

λ

x+

λ¯x

xx+

¯λ¯x

x¯x)

N 21 = −νd0(x

λ+

x¯x

λx+

¯x ¯x¯λx

) − νT 2σλλ¯λ ¯x− ν 3d0 ¯x

λλ¯λ+ T 2σ(λx +

λxx¯x

+¯λx¯x

¯x)

N 22 = d0 + ν 2d0(¯xλλx +

¯x

λ¯λx+

¯x

¯x

λ¯λxx)

+T 2σ(λλx

¯x+

λ¯λx¯x

x¯x+

λ¯λ ¯x

x) + ν 4T 2σ

Where λ = λ(l), ν = ν 0qn, n being the spectral parameter, and σ = ql. The

compatibility condition for this Lax pair produces a series of equations that are

either identities or one of two slightly different copies of qPV. These two copies of

qPV are equal if q¯λ = λ, there are no other restrictions on the parameters. To get

from the form of qPV that comes directly from the Lax pair to the form as listed in

(5.34), we set r = λλ¯λ¯λσ/a0, γ = d0/a0 and T 2 remains as is. This type of condition

on λ is common to every equation that has been calculated by the author. Indeed

it is expected that, when considering a Lax pair with N matrix truncated at A j , λ

must satisfy qλ(l + j − 1) = λ(l) and r = σa0

jh=0 λ(l + h).

We also point out that if one were only interested in this Lax pair and equa-

tion, the coefficients α3

j with j > 4 would be superfluous, they are written abovebecause they are required to find subsequent Lax pairs and equations listed below.

107



Continuing to the next level in the hierarchy, we obtain the coefficients:

α40 = −1/(λλ¯λ

¯λX 1X 1)

α45 = λ +¯λX 1

+

¯

λX 1X 1+

¯

λX 1X 1X 1

α46 = λ2 +

λλ

X 1+

λ¯λX 1X 1

+λ

¯λX 1

X 1X 1+ λ2 + λ¯λX 1 +

λ¯λX 1

X 1+ ¯λ2 +

¯λ¯λ

X 1+

¯λ2

Ceasing at A5 yields a new Lax pair for a fourth order equation also written in [2].

The associated equation, given below, is a reduction of the LMKDV equation under

x =4x. The notation

4x = xl+4 is used instead of ¯x because too many bars become

difficult to read.

¯yyyyy =1 − T 2ryyy

γ yyy − T 2r

where y = ¯x/¯x, log r = γ 0 + γ 1il + γ 2(−1)l + γ 3(−i)l − ql/4 and γ ι = constant.

The relationship between r and quantities in the Lax pair for this equation is r =

λλ¯λ¯λ4

λσ/a0 and q4

λ = λ to ensure compatibility. The L matrix in the Lax pair, as

always, is as in (5.35) and components of the N matrix in the Lax pair are

N 11 = a0 + ν 2

a0(

x

λλ¯x +

xx

λ¯λ¯x ¯x +

x¯x

λ¯λ ¯x4x +

x

λ¯λ ¯x +

x¯x

λ¯λ ¯x4x +

¯x

¯λ¯λ4x)

−ν 2T 2σ(λλ¯λ¯x

x+ λλ

¯λ

¯x ¯x

x4x

+ λ¯λ¯λ

x¯x

x4x

+ λ¯λ¯λ

x4x

)

ν 4a0x

λλ¯λ¯λ4x− ν 4T 2σ(

λx

x+

λx¯x

+¯λ ¯x¯x

+¯λ ¯x4x

)

N 12 = −νa0(1

λx+

x

λx¯x+

x¯λ¯x ¯x

+x

¯λ ¯x

4x

) + νT 2σλλ¯λ¯λ

14x

−ν 3a0( 1λλ¯λ ¯x

+ ¯xλλ

¯λ ¯x

4x

+ ¯xλ¯λ

¯λx

4x

+ xλ¯λ

¯λx

4x

)

+ν 3T 2σ(λλ¯x

+λ¯λ ¯x

x¯x+

λ¯λ ¯x

x4x

+λ¯λ ¯x

xx+

λ¯λ¯x¯x

xx4x

+¯λ

¯λ¯x

x4x

) +ν 5T 2σ

x

108



N 21 = −νd0(x

λ+

x¯x

λx+

¯x ¯x¯λx

+¯x4x

¯λx

) + νT 2σλλ¯λ¯λ4x

−ν 3d0(

¯x

λλ¯λ

+¯x4x

λλ¯λ¯x

+x4x

λ¯λ¯λ¯x

+x4x

λ¯λ¯λx

)

+ν 3T 2σ(λλ¯x +λ¯λx¯x

¯x+

λ¯λx

4x

¯x+

λ¯λxx¯x

+λ

¯λxx

4x

¯x ¯x+

¯λ¯λx

4x

¯x) + ν 5T 2σx

N 22 = d0 + ν 2d0(¯x

λλx+

¯x ¯x

λ¯λxx+

¯x4x

λ¯λx¯x

+¯x

λ¯λx+

¯x4x

λ¯λx¯x

+

4x

¯λ¯λ¯x

)

−ν 2T 2σ(λλ¯λx¯x

+ λλ¯λ

x4x

¯x ¯x+ λ¯λ

¯λ

x4x

x¯x+ λ¯λ

¯λ

4x

x)

ν 4d04x

λλ¯λ¯λx − ν 4

T 2σ(λx

x +λ¯x

x +

¯λ¯x¯x +

¯λ4x

¯x )

We will list one more higher order equation. The next set of coefficients that we

require are

α50 = 1/(λλ¯λ

¯λ4

λX 1X 1)

α56 = λ + λX 1 +

¯λX 1

X 1+

¯λX 1X 1

X 1+

4

λX 1X 1

X 1X 1(5.37a)

Where X 1 = ¯x/x as usual. These coefficients lead to a Lax pair with the L matrix

as before (see (5.35)) and the N matrix below.

N 11 = a0 + ν 2a0(xx

λ4

λ4x5x

+x

λλ¯x+

¯x

¯λ¯λ4x

+x¯x

λ4

λ4x5x

+x

λ¯λ ¯x+

xx

λ¯λ ¯x

4x

+¯x

¯λ4

λ5x

+xx

λ¯λ¯x ¯x+

x¯x

λ¯λ ¯x

4x

+¯x

¯λ4

λ5x

) + ν 2T 2σ(λλ¯λ¯λ

4x

x+ λ¯λ

¯λ4

λx5x

+ λ¯λ¯λ4

λx¯x

x5x

+λλ¯λ4λ ¯x4x

x5x

+ λλ¯λ4λ ¯x ¯x

x5x

) + ν 4a0( x

λλ¯λ¯λ4x

+ x

λ¯λ¯λ4

λ5x

+ xx

λ¯λ¯λ4

λ¯x5x

+x ¯x

λλ¯λ4

λ4x5x

+x ¯x

λλ¯λ4

λ¯x5x

) + ν 4T 2σ(λ4

λx4x

x5x

+¯λ

¯λ4x

¯x+

λ4

λx4x

¯x5x

+λλ¯x

x

+λ¯λx

¯x+

λ¯λx

4x

x4x

+λ

¯λx

4x

¯x¯x+

λ¯λx¯x

x ¯x+

¯λ4

λ ¯x5x

+¯λ4

λ ¯x4x

¯x5x

) + ν 6T 2σ

109



N 12 = −νa0(1

λx+

x

λx¯x+

x¯λ¯x ¯x

+x

¯λ ¯x

4x

+x

4

λ4x5x

) − νT 2σλλ¯λ¯λ4

λx5x

−ν 3a0(x

¯λ

¯

λ

¯

λx

4

x

+x

¯λ

¯λ

4

λ¯x

5

x

+¯x

λ

¯

λ

¯

λx

4

x

+¯x

λλ

4

λ

4

x

5

x

+1

λλ¯λ ¯x+

¯x

λ¯λ

¯

λ¯x

4

x

+¯x

λ¯λ4

λx5x

x ¯x

λ¯λ4

λx4x5x

+¯x ¯x

λ¯λ4

λx4x5x

+x ¯x

λ¯λ4

λx¯x5x

) − ν 3T 2σ(λ¯λ

¯λ4x

xx+

λ¯λ¯λ4x

x¯x

+λλ

4

λ4x

¯x5x

+¯λ

¯λ4

λ¯x

x5x

+λλ¯λ

¯x+

λλ¯λ4x

¯x ¯x+

λ¯λ4

λ ¯x

x5x

+λ¯λ

4

λ ¯x4x

xx5x

+λ¯λ

4

λ ¯x4x

x¯x5x

+λ

¯λ4

λ¯x ¯x

xx5x

)

− ν 5a0

λλ¯λ¯λ4

λ5x

− ν 5T 2σ(λ

x+

λ¯x

xx+

¯λ¯x

x ¯x+

¯λ4x

x ¯x+

4

λ4x

x5x

)

N 21 = −νd0(x

λ+

x¯x

λx+

¯x ¯x¯λx

+¯x4x

¯λx

+

4x5x

4

λx

) − νT 2σλλ¯λ¯λ4

λ

5x

x

−ν 3d0(x4x

λ¯λ¯λx

+¯x5x

¯λ¯λ4

λx

+x4x

λ¯λ¯λ¯x

+

4x5x

λλ4

λ¯x

+¯x

λλ¯λ+

¯x4x

λλ¯λ¯x

+x5x

λ¯λ4

λ ¯x

+x4x5x

λ¯λ4

λx ¯x

+x4x5x

λ¯λ4

λ¯x ¯x

+x¯x

5x

λ¯λ4

λx ¯x

) − ν 3T 2σ(λ¯λ

¯λxx4x

+λ¯λ

¯λx¯x4x

+λ¯λ

4

λ¯x

5

x4x

+¯λ

¯

λ

4

λx

5

x¯x+ λλ¯λ ¯x + λ

¯λ

¯

λ¯x¯¯x4

x+ λ

¯

λ

4

λx

5

x¯x+ λ

¯λ

4

λxx

5

x¯x4x

+ λ¯λ

4

λx¯x

5

x¯x4x

+λ

¯λ4

λxx5x

¯x ¯x) − ν 5d0

5x

λλ¯λ¯λ4

λ

− ν 5T 2σ(λx +λxx

¯x+

¯λx ¯x¯x

+¯λx ¯x4x

+

4

λx5x

4x

)

110



N 22 = d0 + ν 2d0(

4x5x

λ4

λxx

+¯x

λ¯λx+

4x

¯λ¯λ¯x

+

4x5x

λ4

λx¯x

+¯x

λ¯λx+

¯x4x

λ¯λxx

+

5x

¯λ4

λ ¯x

+¯x ¯x

λ¯λxx+

¯x4x

λ¯λx¯x+

5x

¯λ4λ ¯x

) + ν 2T 2σ(λλ¯λ¯λ

x

4x+ λ¯λ

¯λ4

λ

5x

x+ λ¯λ

¯λ4

λx5x

x¯x

+λλ¯λ4

λx5x

¯x4x

+ λλ¯λ4

λx5x

¯x ¯x) + ν 4d0(

4x

λλ¯λ¯λx

+

5x

λ¯λ¯λ4

λx

+¯x5x

λ¯λ¯λ4

λxx

+

4x5x

λλ¯λ4

λx ¯x

+¯x5x

λλ¯λ4

λx ¯x

) + ν 4T 2σ(λ4

λx5x

x4x

+¯λ

¯λ¯x4x

+λ4

λ¯x5x

x4x

+λλx

¯x+

λ¯λ ¯x

x+

λ¯λx ¯x

x4x

+λ

¯λ¯x ¯x

x4x

+λ¯λx ¯x

x¯x+

¯λ4

λ5x

¯x+

¯λ4

λ¯x5x

¯x4x

) + ν 6T 2σ

Incredibly this cumbersome Lax pair has as its compatibility condition the following,

rather simple, fourth order equation

ww =1

w

1 + T 2rww

γww + T 2r(5.38)

where w =4x/¯x and log r = −ql/5+k0+k1 j

l5+k2 j

2l5 +k3 j

l35 +k4 j

4l5 , with ki = constant,

j5 = 11/5, and T 2 is an arbitrary, period-two function of l.

5.4.2 Equations corresponding to reductions of the type

xl,m+1 = 1/xl+d,m

Here we will write down some equations, with their Lax pairs, from the hierarchy

that arises from the LMKdV equation via reductions of the type xl,m+1 = 1/xl+d,m,

d = constant. The procedure used to obtain these results is just the same as that

explained in section 5.3. However, as outlined in section 5.2.2, now A1 = k1xx,k1 = constant, is an end point of the series of terms in the Lax pairs, and the odd

and even powers of ν have been redistributed. We will not list the coefficients used

in finding the results presented here because they are easily obtained from those

used in section 5.4.1. To find alk j as required with the present hierarchy, use αk−1 j−1

from section 5.4.1 and replaceh

X i → 1/h

X i.

111



The first non-trivial equation in this part of the hierarchy is qPIII:

x¯x =1 + T 2rx2

γ x2 + T 2r(5.39)

which was found with the following Lax pair in [2], except here the equation has two

extra free parameters coming from the T 2 term which is an arbitrary, period-two

function of l, and γ = constant. The Lax pair has L as in (5.35) and

N =

ν 2(k1xxλ

+ λT 2σ xx

) νk1x + ν 3λT 2σx

ν k2x

+ T 2σx ν 2( k2λxx

+ λT 2σ xx

)

(5.40)

and λ = qλ for compatibility so r = λλσ/β 2 = γ 0q−l with γ 0 = constant.

The next equation in the hierarchy is an alternative qP II. After setting y = ¯xx

yy = y1 − T 2ry

γy − T 2r(5.41)

The N matrix of the Lax pair for this equation is

N 11 = ν 2[k1x(x

λ+

¯x

λ) − λλT 2σ

¯x

x] + ν 4T 2σ

N 12 = νk1x + ν

3

[k1

x¯x

λλ − (λ +¯λ

¯x

x )T 2σ]

N 21 = νk2x + ν 3[k2

λλx¯x− (λ + λ

x¯x

)T 2σ]

N 22 = ν 2[k1x

(1

λx+

1

λ¯x) − λλT 2σ

x¯x

] + ν 4T 2σ

where ki are constant and, to ensure compatibility, λ = q ¯λ. We set r = λλ¯λσ/k2

causing log r = k0 + k3(−1)l − ql/2, since σ = ql.

The final Lax pair that will be presented from this part of the hierarchy is for

the fourth order equation

x4x =

1 + T 2rx¯x

γ x¯x + T 2r(5.42)

Where γ = k1/k2 = constant and r = λλ¯λ¯λσ/k2. The Lax pair for this equation

112



has the same L matrix again (5.35) and the components of the N matrix are

N 11 = −ν 2[k1(xx

λ+

x¯x

λ+

¯x ¯x¯λ

) + λλ¯λT 2σ¯x

x]− ν 4[k1

x ¯x

λλ¯λ+ (λ

x

x+ λ

x¯x

+ ¯λ¯x¯x

)T 2σ]

N 12 = νk1x + ν 3[k1( ¯xλλ

+ x ¯xλ¯λx

+ ¯x ¯xλ¯λx

) + ( λλ¯x

+ λ¯λ ¯x

x¯x+ λ

¯λ ¯x

xx)T 2σ] + ν 5T 2σ

x

N 21 = ν k2x

+ ν 3[k2(1

λλ¯x+

x

λ¯λx ¯x+

x

λ¯λ¯x ¯x) + (λλ¯x +

λ¯λx¯x¯x

+λ¯λxx

¯x)T 2σ] + ν 5T 2σx

N 22 = −ν 2[k2(1

λxx+

1

λx¯x+

1¯λ¯x ¯x

) + λλ¯λT 2σx¯x

]− ν 4[k2

λλ¯λx¯x+ (

λx

x+

λ¯x

x+

¯λ¯x¯x

)T 2σ]

With this member of the hierarchy we require λ = q¯λ for compatibility, which causes

log r = k0 + k3 jl3 + k4 j

2l3 − ql/3, where kι are constant.

5.5 Discussion

In this chapter we have presented two new hierarchies of nonlinear q-difference equa-

tions, one of which includes qPII and qPV, the other of which includes qPIII in ad-

dition to higher order equations. The relationship between the equations in each

hierarchy was found using a series of Lax pairs and, as such, a Lax pair accompanieseach equation in the hierarchy. All of the resulting equations are non-autonomous

and contain multiple free parameters while each Lax pair is 2 × 2.

Even though these Lax pairs increase in complexity at each level of the hierarchy,

the equations retain the same simple structure while increasing in order and the

number of free parameters. The persistence of a simple structure in the equations

may facilitate the discovery of special solutions applicable to all members of the

hierarchy.

We must point out that some key features of the method used to establish the

hierarchy have not been proven in generality. We simply conjecture their validity

based on agreement with results.

We note that these hierarchies have their roots in reductions from the lattice

113



modified KdV equation, it remains to be seen whether similar results lie behind other

partial difference equations. It would eventually be interesting to find reductions

from lattice equations to the q-Garnier hierarchy constructed by Sakai in [35].

At this point there is still a significant deficiency in knowledge about the generic

solutions of q-Painleve equations. The author is unaware of any instances where

Birkhoff’s theory of linear q-difference equations has been applied to deduce in-

formation about the solutions of q-Painleve equations. The question of the global

properties of solutions remains completely open.

114



Chapter 6

Some future directions: true and

false Lax pairs and equivalent

classes of equations

In this final chapter we explore how false Lax pairs can arise and how they can be

identified by investigating their general forms. We also demonstrate an equivalencewithin some sets of nonlinear equations, by examining parameterizations thereof.

6.1 False Lax pairs

The existence of false Lax pairs is well known [116, 117, 118]. False Lax pairs are

those whose compatibility condition appears to be an interesting nonlinear equation,

but are nevertheless unable to provide any information about the solutions of that

nonlinear equation. For example, take the following linear system:

θ(l + 1, n) = Lθ(l, n),

θ(l, n + 1) = N θ(l, n).

115



Where the Lax matrices take the form

L =

ν [(1 + k)(1 − x) − x(1−x)x(1−x) ] σ−1(1 − x)(1 − x)

σ−1 ν x/x

, (6.2a)

N =

νσ 1 − x

1/(1 − x) νkσ

, (6.2b)

where n plays the role of the spectral variable and ν = qn, σ = ql, k is a constant

parameter, x = x(l) and x = x(l + 1). It is easy to show that the compatibility

condition

L(l, n + 1)N (l, n) = N (l + 1, n)L(l, n) (6.3)

leads to the following equation, without any conditions on the parameter k

x = kx(1 − x) (6.4)

Of course, (6.4) is the logistic equation, possibly the most famous and well studied

chaotic equation known. It is believed that no chaos can exist in any equation that

is solvable through a Lax pair. Therefore, the association between equation (6.4)

and the linear systems in (6.2) is suspected to be fake in some way and warrantsfurther investigation. The question of what might render this association fake is a

difficult one that has had attention in the continuous domain [119] but not so much

attention in the discrete domain. To explain the problem with this Lax pair, we

consider a similar linear problem with general entries in the L and M matrices.

L =

νa b

c νd

, (6.5a)

N = να β

γ νδ

(6.5b)

where ν depends on the spectral variable and all other quantities are functions

of l alone. The compatibility condition (6.3) then yields the following system of

116



equations.

qaα = aα (6.6a)

bγ = c¯β (6.6b)

qdδ = dδ (6.6c)

cβ = bγ (6.6d)

qaβ + bδ = bα + dβ (6.6e)

qdγ + cα = aγ + cδ (6.6f)

Clearly, equations (6.6a) to (6.6d) yield

α = α0σ

δ = δ0σ

γ = γ 0/β

b =c

γ 0β β

where α0, β 0, γ 0 = constant and σ = ql. Substituting these values in equations (6.6e

and (6.6f) brings us to the final two equations.

qdγ 0/β + α0σc = γ 0a/β + qδ0cσ (6.8a)

qdβ +α0

γ 0σcβ β =

qδ0γ 0

cβ βσ + aβ (6.8b)

add (6.8a) and (6.8b) and solve the result for a to get

a =σ

γ 0(α0 + δ0)cβ − dβ/β (6.9)

Finally we can put this expression for a back into either (6.8a) or (6.8b) to get the

equation that is the compatibility condition for the linear system we are analysing.

δ0γ 0

σcβ = d (6.10)

There are no conditions on, or relationships between, the quantities in (6.10) that

come naturally out of the linear system itself, meaning that the system is underde-

termined. Anything we add from this point is artificial and completely arbitrary.

117



To obtain the logistic equation, and corresponding false Lax pair, we set

γ 0 = 1

δ0 = λ

c = σ−1

d = x/x

β = 1 − x

All the other quantities in the Lax pair are given by (6.6).

The key feature in identifying this as a false Lax pair is that there is too much

freedom in the system. No information is given about the evolution of the quantities

in (6.10) and so we cannot gain any information about the solution of the logistic

equation, nor any other equation, from this linear system. This is proof of the

falsehood of the Lax pair considered because the alternative, that this is a real Lax

pair, implies every difference equation is integrable, which is certainly absurd.

There are examples of this type of false Lax pairs that appear in the literature,

claimed to be real. For example, Papageorgiou, Nijhoff, Grammaticos and Ramani[47] presented the following Lax pair for qPI

L =

0 0 x qx

0 xσ(x+1)

0 0

0 0 1/x q/x

ν 0 0 0

N =

0 0 σ/x 0

0 0 x qx

νx 0 1 q

0 ν σqx

0 0

This Lax pair has dPI as its compatibility condition, which is xx = qσ2(x + 1)/x2.

However, beginning with the general Lax matrices of this form, we see that the

118



compatibility condition is a much simpler equation, this fact is illustrated below.

L =

0 0 a b

0 c 0 0

0 0 d f

ν 0 0 0

N =

0 0 α 0

0 0 β γ

νδ 0 1 q

0 νξ 0 0

Where ν = ν (n) and n is the spectral parameter, q is a constant and everything else

is a function of l only. Imposing compatibility (6.3) on this linear system leads to

the following conditions.

aβ = bα (6.11a)

aγ = cα (6.11b)

δ = γ (6.11c)

bβ = 1 (6.11d)

cβ = q (6.11e)

bδ = q (6.11f)

cξ = aδ (6.11g)

c = qb (6.11h)

ξ = qα (6.11i)

This is a simple set of equations to solve, we only write everything out to show

that there are no shortcuts taken and that the solution we obtain is the only one

available. There are numerous ways to proceed, all leading to the same outcome,

we choose to save equations (6.11a), (6.11b) and (6.11g) until last, solving all the

119



other equations for the relevant quantities in terms of a, α and b. This yields

ξ = qα c = qb

δ = q/b β = 1/b (6.12)

γ = q/b

With these values (6.11a) and (6.11b) are identical so we are left with just two

equations

a = αbb (6.13)

a = qαbb (6.14)

which can be reconciled by setting α =√

σb whereupon each of (6.13) and (6.14)

become

a =√

αbbb (6.15)

again, there are no further conditions on a or b that come out of the compatibility of

the two linear systems. In [47] these quantities were set to a = x√σ(x+1)

, b = 1/x so

that (6.15) yields qPI, but one could also set a = λ√

σbb2(1

−b) to find yet another

false Lax pair for the logistic equation. Any other equation can also be represented

by this fake Lax pair using the appropriate choices of a and b.

We remark that these two linear systems do in fact form a true Lax pair, albeit

for a trivial equation. This can be seen from by canceling a in (6.13) and (6.14)

which, after substituting y = α/β , yields

y = qy

This is the only evolution equation that arises naturally from the system itself. It is

therefore concluded that the only equations for which solutions can be reconstructed

from the monodromy data, is this simple equation and its transformations and

reductions.

120



One further example, this time of a false Lax pair for a P∆E, was worked through

in section 3.4.3. The Lax pair identified there takes the form

L = a F 1 b

c F 1F 2 d F 2 , (6.16a)

M =

α F 2 β

γ F 1F 2 δ F 1

. (6.16b)

Where lower case letters depend on the independent lattice variables l and m, and

the upper case F i depend on the spectral variable such that F 1 = kF 2, k constant.

In section 3.4.3 it was explained that the evolution equation for the above Lax pair

is simply the following, underdetermined set of equations:

d = −bα/β, (6.17a)

γ = −aα/b, (6.17b)

δ = −aβ/b, (6.17c)

c = −aα/β. (6.17d)

This underdetermined set of equations does not properly define a nonlinear system,which is why the Lax pair is false, but could mistakenly be imbued with meaning

under certain choices of variables. For example, setting

α = −β

bf (d),

while keeping d as the dependent variable and allowing equations (6.17) to define γ ,

δ and c, leads to the following Lax pair

L =

a F 1 ba

bf (d) F 1F 2 d F 2

,

M =

−β

bf (d) F 2 β

aβ

bbf (d) F 1F 2 − a

bβ F 1

.

121



It is easy to check that this Lax pair has as its evolution equation:

d = f (d),

where f (d) is an arbitrary function of d and any iterations thereof.

6.2 Equivalent evolution equations

In this section we describe how some sets of well known nonlinear integrable P∆Es

can actually be thought of as one equivalent system.

6.2.1 Common partial difference equations

The motivation for this section lies with the following conundrum: Lax pairs with

the following spectral dependence

L =

a b

ν c d

, (6.18a)

M =

α β

ν γ δ

, (6.18b)

where ν is the spectral variable and all other terms are lattice terms, fall into the

class that have LMKdV2 as their evolution equation. However, in [37], Lax pairs

with exactly this spectral dependence were shown to be associated with the cross

ratio equation,

(x − x)(x− ˆx)(x − ˆx)(x− x)

= λ21

µ21

, (6.19)

where λi = λi(l) and µi = µi(m) are arbitrary throughout this chapter. This

suggests a transformation between LMKdV and the cross ratio equation that the

Lax pair itself should be able to elucidate (possibly the known Miura transformation

[78]). During a study of the Lax pair for the cross ratio equation, it was noticed that

122



the set equations from the compatibility condition of the general Lax pair in (6.18)

would yield either the cross ratio equation or LMKdV, depending on the order in

which the equations were solved. How can one set of equations, being solved in

a general way without any arbitrary conditions being imposed, yield two different

evolution equations?

The answer is that neither the LMKdV equation or the cross ratio equation has

solved the compatibility condition fully, each of these evolution equations can be

parameterized one more time. To see this, rewrite (6.19) as

λ1(x − x)λ1(x − ˆx) = µ1(x − x)µ1(x − ˆx),

which is parameterized for

yy =ρ

λ1(x− x), (6.20a)

yy =σ

µ1(x − x), (6.20b)

where log ρ = (−1)m log λ(l) and log σ = (−1)l log µ(m).

On the other hand, LMKdV often takes the form

ˆy = yy − µ2

λ2y

y − µ2λ2

y,

which can be rewritten forˆyy

µ2− yy

µ2=

ˆyy

λ2− yy

λ2.

In this form, it is clear that the LMKdV equation supports the following parametriza-

tion

yy = λ2(x − x + λ3), (6.21a)

yy = µ2(x− x + µ3), (6.21b)

Comparing equations (6.20) and (6.21) shows immediately that LMKdV and the

cross ratio equation are in fact equivalent. The superficial differences arise through

123



the terms ρ and σ in (6.20) and λ3 and µ3 in (6.21), however, these terms are all

introduced during the process of parametrization and can be set to unity or zero

respectively, thus bringing about the equivalence. In fact, the common forms of

LMKdV and the cross ratio equation do not hold all of the possible ‘parameter

functions’ (the arbitrary functions ρ, σ, la1 and µi that play the role of parameters),

when all of these are included both equations can be parameterized to obtain

yy = ρ4λ4(x − x + λ5), (6.22a)

yy = σ4µ4(x − x + µ5). (6.22b)

Equation (6.22) is a integrable, coupled pair of first order difference equations that

encompasses generalized forms of both LMKdV and the cross ratio equation.

6.2.2 Higher order versions

The full pair of coupled equations referred to as LMKdV2 is,

λ1

σ

ˆx

x

+µ2

ρ

y

y

= λ2σy

y

+ ρµ1

ˆx

x

, (6.23a)

ρµ1x

x+ λ2σ

y

ˆy=

µ2

ρ

y

ˆy+

λ1

σ

x

x. (6.23b)

Where it is understood that the terms in (6.23a) are not necessarily related to

terms using the same notation as in the previous section, although the naming

system is consistent with λi = λi(l), µi = µi(m), log ρ = (−1)mλ3(l) and log σ =

(−1)lµ3(m). This pair of second order nonlinear equations describes the general

dynamics that arise from the compatibility of the two linear systems (6.18). A

further parametrization is achieved by adding the two equations in (6.23a) together

and finding exact differences, although to bring these about we must set ρ and σ to

124



unity. It is not difficult to verify that the following parametrization holds:

λ1x

x= v − w + λ3 − r1, (6.24a)

µ1 xx = v − w + µ3 − s1, (6.24b)

λ2y

y= w − v + λ3 + r1, (6.24c)

µ2y

y= w − v + µ3 + s1, (6.24d)

where v and w are introduced terms dependent on both lattice variables l and m,

r1 = (−1)mλ4(l) and s1 = (−1)lµv(m).

By eliminating x and y from equations (6.24), a pair of coupled, cross ratio typeequations in v and w can be extracted.

( ˆw − v + λ3 − r1)(w − v + µ3 + s1) = ( ˆw − v + µ3 − s1)(w − v + λ3 + r1),

(ˆv − w + λ3 + r1)(v − w + µ3 − s1) = (ˆv − w + µ3 + s1)(v − w + λ3 − r1).

6.3 Discussion

In this chapter, we explored different ways in which false Lax pairs can occur and

how to find equivalences between equations by considering their parameterizations.

In [38] it was shown (see equation (2.6) in that paper) that the Adler system, also

known as Q4, can be written in the form

f (x)f (x) + g(x)g(x) = k, (6.25)

where k is a constant. It will be interesting to see what equations Q4 is equivalent

to.

125



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